Beginner  Geometry

Steps to develop practices, or skills and concepts, with minimal theory, or very short chains of reason.  That is, prerequisites for each practice are minimal.  Each practice may appear to be obvious to students, so that minimal justification is needed after their introduction. 

Steps or Topics

1. Geometry and Formula Evaluation - a help for algebra.
2. Two Definitions of Similarity (Optional Reading)
3. Geometry in or with maps, plans and designs -using similarity in map or drawing form to find missing distances and angles. 
4. Using Similarity in Equation form to find missing lengths and angles of triangles and (?) polygons - latter optional. 
5. Right Triangle Trigonometry upto law of sines -using similarity in trig disguise.
6. Length, Areas and/Or Volume Scale Factors Backwards and Forwards
    for Maps, Plans and Models in 2 and 3D - read the two definition of similarity if you skipped them before this item. 
7. Rectangular and Polar Coordinates with unsigned and then signed numbers -
8. More Geometry with maps - Navigation and Orienteering. 

To learn more see the site area on Euclidean-Geometry To Complex No.s, and the next page 5. More Geometry.

Step 1: Geometry and Formula Evaluation,
a step common to both geometry & algebra

The algebraic description of length and areas of triangles, squares, rectangles, trapezoids, parallelograms, circles and fractions of circles provides formulas for student to evaluate.   Detail formatting rules for the evaluation of geometric formulas, diagram drawing and labeling included,  show students how to show work - how to communicate the setting, the steps in their reasoning and results in the evaluation of geometric formulas in an observable and correctable manner on paper.  That is a performance objective easily understood and met.. 

The use or role of letters or more generally symbols as placeholders in formulas and identifiers (labels) on diagrams provides a starting point for algebraic ways of writing and reasoning in general where letters or symbols or expression are placeholders for numbers and quantities without an immediate geometric significance.

Examples:

1. Give Formula Evaluation Exercises for areas of squares, rectangles,  triangles, parallelograms and circles with justification where possible of all except for the formula for the area of the circle. That latter requires calculus (or a numerical study of how the area of of circles is proportional to the square of the radius).
   
2. Give Formula Evaluation Exercises for perimeters of squares, rectangles, circles and semicircles, triangles, parallelograms, regular polygons.  justification where possible of all except for the formula for the area of the circle. The justification of the circle perimeter formula  requires calculus (or a numerical study of how the perimeter of a circle is proportional to its radius).

Teachable Moment (painful): Recognition that multiplying by a half gives the same result as dividing by a half is an example of the notion that different formulas when evaluated will give the same result, or in brief the notion that two different expression may be equal or have the same value.  The idea for this come from a student painful objection to my writing two formulas for the area of triangle- one using the factor one half and the other using division by two. I was not being consistent. Consistent use of one or the other formula might have avoided the issue.

2 Two Definitions of Similarity 

Similarity theory in high school geometry may employ the following definition as is.

Definition A: Two planar polygonal figures (triangles) are similar when and only when (i) corresponding angles are equal (have the same measure) and (ii) corresponding sides are proportional.

  The latter codifies the notion of two planar regions or curves in plane having the same shape, incompletely as only polygonal curves and regions are considered, but that is good enough for the further needs of high school mathematics. The further study of trigonometry with right triangles  is based on the similarity criteria for  triangles.   Definition (A) limited to triangles or  right triangles -  would be sufficient for  trigonometry with right triangles. The general theory of similarity need not be discussed for the sake of right triangle trigonometry. 

The UK English national curriculum emphasizes all circles are similar and so are all squares. Circles are not polygons. So they are beyond the reach of statement (A).  Squares are beyond the reach of statement (A) when the polygons are limited to triangles. 

However, like Shapes are met and discussed in  Elementary School.  In reading and writing letters and in seeing objects as they exist or on paper, we recognize letters and objects which have the same shape or nearly the same shape but different sizes. Size but not shape varies as we move to or away from the letters or object. Size thus depends on distance. The geometric theory of optics says or suggest how.   Like shapes are recognized not only in reading and writing, but also in maps, drawings and picture - most likely with distortions ignored. In practice we ignore small differences in declaring two shapes or letters to be alike. 

More generally, in planar maps and plans, we recognize when corresponding sets or geometric figures in reality or a map have the same shape, are alike or similarly.   In essence, the number of unit distances in a straight line between corresponding points  in reality or different maps is invariant - the same for all maps and reality. In addition, corresponding angles are equal. Otherwise a distortion is sense or seen. If we were too digitalize an object in reality or as represented in a map with respect to a coordinate system or corresponding systems in reality and in the maps, we would get a single set of order pairs.  There-lies an invariance. That invariance implies or suggest the following coordinate or analytic definition of similarity:

Definition B: Two Planar figures are similar when and only when there are represented by the same set of coordinates in a pair of coordinate systems, one for each object.  

In the foregoing definition, the coordinate system could 1D, 2D or 3D etc. 

Example 1 -  Arcs and Sectors of Circles: Arcs and sectors of circles can be compared using their central angles and the radius of the circles containing the arc.  Two arcs or sectors are isometric if their radii and central angles are equal.  Two arcs or sectors are similar if their central angles are equal.  Direct Measurement as well as map drawing  implies corresponding lengths in similar arcs are proportional.

Example 2 - Rectangles - All rectangles with the same aspect ratio, that is the same height to base ratio are similar.  In consequence, all squares are similar. 

Example 3 - Construction from Drawings or Plans:  An plan draws an object to scale. The initial scale has one value. Another scale is then used as well. Construction of the object to two different scale then results in a pair of similar objects (1D, 2D or 3D). 

Definition A is insufficient to talk about similarity of arcs and sectors of circles, 2D objects with curve  boundaries, and solid or 3D objects in general. 

We need to recognize that in practice, high school geometry employ two definitions of similarity.  

Exercise (i) for Students in Honours Mathematics Programs, University Level:  Show that if two  polygons with corresponding sides and angles are similar by definition (A) then then the set of points formed by the sides of this polygons are similar in accordance with definition (B).  Use the uniqueness of solutions to differential equations to show this.   Additional Exercise (ii): Show the foregoing for the interiors (respectively exteriors) of the polygons as well - an exercise in topology connected with the simple closed curved theorem. It should be possible, but I do not if it is trivial or not.

3: Geometry with Maps, Plans and Designs 
(Using Similarity in Drawing or Map Form)

Geometry began the art of measurement on land or in the plane. That measurement can be done directly with surveying instruments, tape measures and angle measuring devices included. Superposition of line segments intersecting at interior and/or points allows a comparison of angles and provides a basis for measurement of angles using protractors in terms of degrees. Length and angle measurement can be done indirectly with maps and plans drawn to scale. When the scale is identical in all directions, drawn angles equal the actual angles (real or intended) but actual (absolute) lengths are proportional. The scale factor is or gives the proportionality constant for all lengths. The scale factor gives the ratio of actual and drawn unit length in reality and on the map or plan.  The concept of similarity is implicit here.

Maps, plans, designs and drawings made to scales less than, equal or greater than 1  may be used for locating objects and for describing movements along trails or paths, actual or intended.   

1. Maps: In maps drawn or redrawn, the image of a straight line segments and circular arcs are also straight line segments and circular arcs.   Whence the images of figures made of straight line segments and circular arcs are also made of straight line segments and circular arcs. Image element are seen to be proportional to their pre-images in the original figures.  All the foregoing can be shown or implied by many examples, and then assumed as a drawing and design shortcut or tool.
   
   There is an innate ability to recognize like shapes, close-up and far-way, within the level resolution capabilities of eyes - a level that may vary. The ability to read and write letters, digits and further symbols, and to recognize (read) and draw line segments,  squares, circles and semi-circles depends on that ability.  The abiltiy to recognize shapes and figures in pictures and diagrams  also depends on this ability.  Primary students and teachers learning to read and write, and learning geometry, may recognize like or similar shapes without any mention of the formal characterization of similarity that appears say in secondary school mathematics. Geometric optics suggest two figures, polygonal or not, in different maps have the same shape if one is the projection or scale drawing of the other  - undistorted.  Distortions would follow from different scales on different axes. The secondary level discussion and definition  of similarity of polygons and circles in a single plane or appearing on different maps characterizes and codifies similarity in terms of corresponding angles being equal and corresponding lengths being proportional formalizes or codifies that innate ability but not fully as the geometric optics projection, perspective geometry and/or scale drawing viewpoint.    The equivalence of the latter to the primary school identification of geometric figures and curves having like or same shapes is incomplete as the formal discussion only involves polygonal figures.  
   
2. Map Drawing or Construction:  In drawing maps of physical situations and objects or points there-in,, students may determine the image of an object or map in the map by using physical measurements to determine the location of the point relative the bottom-left corner of the map with the aid of real-life unsigned rectangular and/or polar coordinates. For example, students may be asked to draw or map to scale, their current classroom and the location of key objects there-in -  desk and chairs, blackboards, doors, windows, etc.  Line segments, squares and rectangles, and part of circles, may be used to depict the latter objects on the map.  Desk should be drawn in proportion - so that aspect ratio of their sides (top view) is maintained.  Teachers could introduce four objects  with a triangular top view in the classroom and get students to plot them in a room map or plan with the aid of (i) three vertex coordinates,  (ii)  the coordinates of the end points of one side (top view) and the use of  the SSS, SAS and ASA physical measures to draw the images of the objects (triangular top view) in the map.  Division of the room and map into corresponding grids may help.
   
   Show students that drawings with different scales in different directions distort angles and lengths.  
   
3. Measure or Calculate Distances and areas with Maps and Plans.  Students may measure the drawn, on-map distance between two points on a map using a ruler or a tape measure, and then determine the pre-image points with by multiplying by a scale factor (proportionality constant).   Let the unit length in the map be the image of an actual or real-life unit length.  Then map unit square is the image of the actual or real-life unit square. Simple examples may imply that measure relative to the unit lengths and areas are invariant - that is the same in the map and in actuality. Whence lengths and areas of a figure or its map image can be measured or calculated relative to unit length and area on the map or in real life.  The advantage of maps, plans and drawing in calculating lengths and measures, and in route planning, appears when the actual or real life actual (absolute) measures are not feasible.  In other words, maps, drawing and plans provide a means for the indirect measurement as relative lengths and areas are invariant. Whence on-map (on drawing or on-plan) measurements provide an alternative to real or actual measurements.   For surveying and navigation, information that is sufficient to draw a length or figure to scale allows the missing dimensions and areas in the figure to be determined from the drawing.  This exercise needs to be done years before the study of trig begins.  
   
   The foregoing may be done before the use of coordinates and then after. See the introduction of coordinates below.
   
   Example: The Angle Side Angle method can be employed by a ship navigator to locate a his location from the bearing of two landmarks.  
   
   Working With Maps and Plans.  Planar triangles can be drawn and duplicated from side and angle data, namely the lengths of three sides, from the lengths of two sides and the measure of included angle, and from the measures of two angles and the included length. Planar triangles can be drawn to scale from side and angle data, namely the lengths of three sides, from the lengths of two sides and the measure of included angle, and from the measures of two angles and the included length. Drawing to scale gives a similar triangle. Once the triangle is drawn and duplicated in full or to scale, missing lengths and angles can be measured and thus found from the drawing or image of a actual or intended triangle. The triangle of interest may be in a horizontal, vertical or slanted plane

4.  Solving for Missing Lengths in Triangles using Similarity in Equation Form

In step 2, students learn how to solve triangles via counting units or measurement on scale drawings - the same scale in two orthogonal directions.  In this step,

 In this step, we may assume the following definition (A) as is or in the special case where the polygons are required to be triangles. 

Definition A: Two planar polygonal figures (triangles) are similar when and only when (i) corresponding angles are equal (have the same measure) and (ii) corresponding sides are proportional

Assumption A: If two corresponding angles in a pair of triangles are similar, then the triangles are similar.  

Assumption B: If corresponding in a pair of triangles are proportional, then the triangles are similar.  

Exercises  and examples on recognizing similar triangles using definition (A) or assumption (A) and (B)  are now needed to test comprehension of them.  More exercises on the forward and backward use of similarity relations are then required.

See  the site coverage of similar triangles for more information. 

5. Right Triangle Trigonometry

Similarity theory in geometry says when two polygonal figures have the same shape. Two planar polygonal figures (triangles) are similar when and only when (i) corresponding angles are equal (have the same measure) and (ii) corresponding sides are proportional. The latter codifies the notion of two planar regions or curves in plane having the same shape, incompletely as only polygonal curves and regions are considered, but that is good enough for the further needs of high school mathematics. The further study of trigonometry with right triangles  is based on the similarity of triangles. Similarity theory for right triangles would be sufficient for  trigonometry with right triangles.. 

For angles q  between 0 and 90 degrees, similarity of right triangles implies the trig fractions

are independent of the choice of right triangles used to compute them. if you  replace the unit circle right triangle by a similar right triangle. The latter  formulas for  may be used to compute with any right triangle where sides are labeled opposite and adjacent for an angle  q    The  further trig functions may be defined as follows.

when the divisors are nonzero. Exercise: Express these further trig functions as ratios of the sides opposite, adjacent and/or hypotenuse of the above right triangle.

Trig functions link the ratio of two sides of a right triangle to cosines, sines and tangents of an angle. Knowledge of two sides in right triangle gives knowledge of the third by means of Pythagorean theorem, and of the values of the trig functions for the angles in the triangle.  Computation of unknown side lengths, unknown hypotenuse lengths and unknown angles is useful in land measurement (geo - metry) and also in navigation.

From one-to-one properties of trig functions for angles between 0 and 90 degrees or ½p, one can define (say how to compute) inverse trig functions (more functions on your calculator)  to compute the angles from the ratio of sides. 

The forward use of the above formula calculates trig functions for an acute angle q  from the lengths of sides (legs) of right triangles. A backward use of one of these formulas would determine the length of a leg from the value of  or a trig and the length of a side.  Another backward use would be to find the angle given the value of one of the trig functions directly, or through the calculation of trig fraction. 

Trig functions link the ratio of two sides of a right triangle to cosines, sines and tangents of an angle. Knowledge of two sides in right triangle gives knowledge of the third by means of Pythagorean theorem, and of the values of the trig functions for the angles in the triangle.  Computation of unknown side lengths, unknown hypotenuse lengths and unknown angles is useful in land measurement (geo - metry) and also in navigation.

From one-to-one properties of trig functions for angles between 0 and 90 degrees or ½p, one can define (say how to compute) inverse trig functions (more functions on your calculator)  to compute the angles from the ratio of sides. 

The forward use of the above formula calculates trig functions for an acute angle q  from the lengths of sides (legs) of right triangles. A backward use of one of these formulas would determine the length of a leg from the value of  or a trig and the length of a side.  Another backward use would be to find the angle given the value of one of the trig functions directly, or through the calculation of trig fraction. 

Remark: The cosine law can be left to after a discussion of the unit circle definition or redefinition of trig functions to obtain values for trig functions for angles that are not acute. 

Calculation

One may define trig functions by saying how to compute them in principle as above, but then one computes or approximates them in practice from tables and slide rules (old fashioned approach) or using calculators (the new approach). Unfortunately in this practice,  the tables, slide rules or calculation devices are black boxes which provide results, but whose derivation or justification is not commonly known. This departs from the principle of understanding the computations one does, but the numbers computed by these black boxes can be checked in simple cases involving 30-60-90 right angles (obtained by bisecting angle in an equilateral triangle) and involving 45-45-90 degree triangles obtained by SAS construction using two sides of equal length, and taking 90 degrees to be the included angle.

Remark: Understanding how to do exact arithmetic with whole numbers and square roots to  represent the sines and cosines of 30, 45 and 60 degrees using 30-60-90 and 45-45-90 right triangles is must do in the high school preparation for calculus. 

Background Information: Solving Triangles Using Similarity or Trig Functions:  Show how right triangle trig provides an alternative to similarity analysis in solving triangles. Given a large right triangle with determined (explain why) by SAS with the aid acute angle A and the measure of one its sides (opposite, adjacent or hypotenuse),  a triangle with sides that cannot be measured directly, we can find the lengths of the remaining sides by (i) drawing a similar right triangle and measuring (similarity implies the ratio of any two sides in a figure equals the ratio of the corresponding sides in any similar figure) or (ii) using tabulated or electronic calculator given values of right triangle trigonometry functions cos(A), sin (A) or tan(A) found without immediately drawing a similar right triangle, but found in principle from drawing many right triangles and tabulating the results.  In other words, the use of trig functions hides or buries the use of similarity in solving right triangles with missing lengths in the earlier link of  trig functions values for acute angles to  the ratios of adjacent sides of right triangles.  

Application: The SAS triangle area formula 

The foregoing suggests the following SAS method for calculating the area of a triangle given two sides d and D and the included acute angle q between them.

Area S  = ½ (length of side)(length of another side) sin (included angle)     

Call the latter the SAS triangle area formula.

Extension to the Obtuse Angle Case

- this case could be postponed until trig functions of all angles have been met.

Here  sin q   = h/a and hence h = a sin q  . 
Therefore  

Triangle Area S  =  ½ c h  = ½ c a sin q      

Now put sin b   = sin  q  = sin (180^o - b)
when b  is between 90 and 180 degrees.

The latter  implies 

Triangle Area S  =  ½ c h  = ½ c a sin b 

By putting sin b   = sin  q  = sin (180^o - b)  when b  is between 90 and 180 degrees., the previous formula triangle area  = The latter half the product of two sides times the sine of the included angle.  is extended to  the case where the included angle is obtuse.  

Note:  A triangle is half a kite or parallelogram. So the foregoing also implies SAS formulas for the areas of kites and parallelograms which can be developed in class notes or in exercises. 

Derivation of The sine law:

- this case could be postponed until trig functions of all angles have been met.

From the SAS triangle area formula, the area of  triangle ABC 

is given by  S  = ½ ac sin B where B in an overuse of the symbol B, denotes the angle at vertex B.  In general, there are three formulas

• S  = ½ bc sin A
• S  = ½ ac sin B
• S  = ½ ab sin C

Equality of the first two formulas ½ bc sin A = ½ ac sin B gives   bc sin A =   ac sin B and hence   b  sin A =   a  sin B. The latter in turn (divide both sides by ab) gives 

 Equality of the last formulas ½ ac sin B =  ½ ab sin C likewise gives 

Likewise equality of the first and last formulas give

Now instead of writting (I), (II) and (III) separately, we write the simultaneous equalities

The latter expression is called the sine law, and its a brief form of stating (I), (II) and (III).

Remark: Formula (I) involves four quantities, namely two lengths and two angles.  The backward use of (I) as is, or rewritten as an equality of reciprocals, which ever is most convenient, means that whenever three of the four quantities is given, the fourth can be found.  Formulas (II) and (III) likewise involve four quantities.  They too can be used in a backward manner.

Sine Law Application Ideas:   Use the unit circle introduction of trig functions and reflection about vertical axis to show that sin(180 - A) = sin (A) for acute and obtuse angles A. Next prove the sine law for triangles, scalene or not.  Interpret the sine law in the case of right triangles and imply it works, with some overlap or redundancy [to do:  redundancy to be spelled out.]  Next apply the sine law forwards, backwards and side ways to solve right triangles.  Point out the option of drawing similar triangles and measuring in each way the law is used. 

6: Scale Factors for Maps and Models in 2 and 3D

1. Measuring lengths and areas with relative and actual (absolute) measures: Suppose we take 1 meter to be the unit length for measurement of distance.  Then a  curve or length with actual or actual (absolute) length of 5 meters has a length of 5 relative to the unit length of one meter.  Further a region with actual or actual (absolute) area of 14 square meters, that is, 14m² has area 14 relative to the unit area of m² or 1 square meter.  
   
2. Relative lengths are invariant: Suppose a line segment 8 m (8 meters) long is drawn on a map with a scale of 10 to 1. Then the drawing of the line segment will be 8 dm (8 decimeters) long.  The original unit length, that is one meter, is gives or corresponds to a 1  decimeter unit length.  The original line segment and its image both have relative length 8 with respect to the original unit length (1 meter) and the unit length (1 dm) on the map.  So relative lengths are unchanged.  Likewise, if a sequence of line segments forms a piecewise linear path in the original plane, then the images of the sequence drawn on the map forms the image path, piecewise linear too,  in the map.  Both paths will have different actual (absolute) lengths, but identical lengths relative to the unit length in the physical situation and in the map.  So again, relative lengths are invariant. 
   
   Finally and optionally, if the relative length of a curve in a physical plane is the limit of the lengths of sequence of piecewise linear approximation to it, the original curve,  then the image curve in a map will be the limit the lengths of sequence of corresponding piecewise linear approximation to IT, the image curve, and vice-versa. Whence the image curve and the original curve will have the same lengths.  The piecewise approximation can be taken with zero error (to be exact) on any portion of the curve which is linear. 
   
3. Relative areas are invariant: Suppose a rectangle with dimension 3 meters by 4 meters is drawn on a scale of 1 to 10 on a map. The image is a rectangle of dimensions 3 dm by 4 dm.  The actual (absolute) or actual area of the original square is 12 square meters or 12m²   while the area of the image is 12 square dm or 12 dm².  Observe that the image of a the unit of area, that is a square meter, is the unit of area, a square decimeter, in the map.  Here we see the area 12 of the original rectangle relative to the original unit area equals the area 12 of the its image rectangle on the map with respect to the map unit area = the image of the original unit area.  So area of the rectangular region and its image defined relative to the unit squares  and its image is the same. Likewise,  the areas of square and their images relative to the unit areas in the pre-image and image planes are equal.  The key word here is relative.  
   
   Finally and optionally, areas of regions in the original plane can be approximate relative to the unit square in the original plane by covering the region by small squares and finding the limit in relative or actual (absolute) terms. Do the same in for the image of the region and using the images of those small squares gives the same sequence of approximations for the relative area of the image and hence, in the limit, the image region has the same relative area as its pre-image - the original region. 
   
   Application to Note Taking:  A teacher draws a parallelogram on a board with height of 5 units and a base length of 4 units.  Each note taking student in the class draws a similar parallelogram with height 5 units and base 4 units, but the unit length used in all drawings of the students and their one teacher are not the same. None the less, the students and teacher all see that their version of the parallelogram, the original and all its images, have a common area of 20 = 5 x 4 relative to their unit of measure. In all calculations of area of a figures, figures whose corresponding dimensions relative to a unit length in the diagram or map containing the figure are identical, all have the same relative area.  Whence relative area calculation for a single figure - the original - may done with a figure that is similar to it. We may same for composite figures - figures that can be decomposed or split into smaller similar figures, so that the sum of the areas, actual or relative, equals that of the original composite.   TASK: Say or rewrite the foregoing in a clearer manner. 
   
   Extension: In a like manner, when 3 dimensional objects are designed or mapped, relative lengths, relative surface areas and relative volumes are invariant, that is, equal for each original object and any similar object that models it. 
   
4. Scale Factors K, K² and K³  for actual (absolute) Measures:  In mapping or modeling a 1, 2 or 3 D object or figure, the original unit length corresponds to an image unit length  = K times the original unit length. We take that image unit length to the unit length for the map or model, and thus for the calculation of unit area for 2D regions or surfaces, and for the calculation of unit volume for 3D models of 3D objects.  Whence 
   
   image unit length = K * original unit length 
   image unit area    = (K * original unit length)² =  K² (original unit length)² = K² original unit area
   
   and 
   image unit volume    = (K * original unit length)³ =  K³ (original unit length)³ = K³ original unit volume.
   
   That is
   
   image unit length      = K * original unit length 
   image unit area         = K² original unit area
   
   and 
   image unit volume    =  K³ original unit volume.

   For corresponding lengths, surface areas and/or volumes, the relative measures are equal by previous arguments.  Whence 
   

   image actual (absolute) length = K * original length                    or  L_image  = K L_original
   image actual (absolute) area    = K² original actual (absolute) area            or  A_image  = K² A_original
   
   and 
   image actual (absolute) volume    =  K³ original actual (absolute) volume  or  V_image  = K³ V_original

   Remark 1: The numbers are scale factors or proportionality constants for length, area and volume respectively. When one is calculated or obtainable, then so are the others via arithmetic operations of squaring, cubing, taking square roots and/or taking cube roots.  Moreover, if the ratio of a pair of image and original lengths, areas or volumes is known, then one of the scale factors K, K² and K³ and hence all may be calculated.  See site discussion of forwards and backwards use of formulas and proportionality relations to learn more.
   
   Remark 2:  Memorization of squares and cube roots of 1, 2, 3, 4 and 5 may help in the backward calculation of the scale factors in exercises that develop or encourage forward, backwards and sideways use of the proportionality between lengths, areas and volumes in maps and models, and in real life.
   
   Remark 3: The proportionality factors K, K² and K³ also apply to quantities that are proportional to length, area and volume in the building of models to part, full or oversized scale. Exercise: Explain why. 
   
   Remark 4:  Say the scale of a map or plan is 1 to 10000 then 1 cm on the map corresponds to 100 meters =10000 cm on the map. Now let d = distance measured on the map = n cm. Let D = the corresponding distance in practice (actuality) measure in meters. Then  we expect 1 to 10000 = d to D or  10000 to 1 = D to d. So 10000 = D/d or 10000 d = D.  Now d = n cm gives D = 10000 (n cm) = 100 x n 100 cm = n 100 meters.  Thus the "unit distance" 1 cm on the map corresponds to the unit distance of 100 meters in actuality. 
   

5. Similarity by Design: If two artificial bodies S and S' appear to have the same shape, then it likely but not guaranteed that they come from  the same plan. If they do, they are similar.
   

6. An Alternative to Trig:   Suppose S is a 2D or 3D figure that is similar to another figure S'. Then similarity or proportionality implies the ratio of any two sides, areas or volumes in the figure equals the ratio of the corresponding sides, areas or volumes in any similar figure S'.  If one aspect of  figure S is too large to measure, the construction of a similar figure may make that aspect measurable with the aid of a proportionality constant. 

7: Rectangular and Polar Coordinates

Preparing for the Use of Coordinates and Arrows (to Represent Displacements) on Maps and Plans
Easily Covered Background Information. 

The introduction of rectangular and polar coordinates may be done here, in the development of algebra or with the further discussion of advanced geometry

1. Rectangular Coordinates for Maps with unsigned numbers:  Ordered Pairs [a,b] of Mixed numbers, proper and improper fractions and decimals with square brackets may be introduced as coordinates to locate points on rectangular maps when the origin of this unsigned coordinate system is place at say the bottom-left corner of each map. The introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions - and based on the introduction of a square grid covering the map. Each square in the grid can itself by covered by a grid of smaller squares, and so on, ad infinitum.
   
   Note: the foregoing coordinates [a, b] are relative to the choice of unit length.  actual (absolute) coordinates would use coordinates of the form [A, B] = [a units, b units] with ordered pairs of mixed number multiples of units (quantities). 
   
2. Rectangular Coordinates for maps with Signs:  Ordered Pairs [a,b] of Mixed numbers, proper and improper fractions and decimals with plus and minus signs as prefixes may be introduced as coordinates to locate points on rectangular maps when the origin of this unsigned coordinate system is not placed at the bottom-left corner of each map. As before, the introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions - and based on the introduction of a grid of unit squares covering the map. Each square in the grid can itself by covered by a grid of smaller squares, and so on, ad infinitum. The boundaries of the map need not be aligned with grid elements. Make sure that students are aware that the coordinates of a point are relative to the length of unit vectors. 
   
3. Polar Coordinates with unsigned numbers:  Ordered Pairs (r, q)  of Mixed numbers, proper and improper fractions and decimals with round brackets may be introduced as coordinates to locate points P on rectangular maps when the origin of this unsigned coordinate system is place at say the bottom-left corner of each map. The introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions. Here r  = the distance of the point P from the origin while q = angle of the ray from the origin to the point P. The angle would be between 0 and 90 degrees for points in the first quadrant, and between 90 and 360 degrees for points in other quadrants.
   
   Note: the foregoing coordinates (r, q) are partially relative to the choice of unit length for distance and actual (absolute) for degree measure.  actual (absolute) coordinates would use polar coordinates of the form (R, q) = (r units, q )  with R being the actual (absolute) quantity r units, and q (still) being the actual (absolute) degree measure of angle.
   
   Note:  The angle q of a point is determined modulo 360 degrees.  One might speak of the angle, modulo 360 degrees, for the sake of having a "unique angle".  That angle might be identified with a point on a unit circle. 
   
   Remark for Items 1, 2 and 3 above:  It can be implied or suggested in Euclidean Geometry that rectangular coordinates [a,b] of point in a rectangular region of the plane which includes the origin [0,0]  uniquely determine a right triangle with one leg on the horizontal axis and a hypotenuse given by the straight line segment between the origin [0,0] and point [a,b]. Then the length of the hypotenuse gives R = r units while the standard angle it makes with the horizontal axis gives angle q.   Conversely, the polar coordinates of a point determines the legs of a right triangle with one leg on the horizontal axis.  The lengths of those legs and the identification of the quadrant in which point lies determines the points rectangular coordinates [a,b]. 
   
4. Map Mastery Exercises: Student mastery of rectangular and polar  coordinates may be developed and verified by exercises which require them to locate and plot individual points (dots) from point coordinates.  Student comprehension of rectangular coordinates may be further developed and verified by exercises which require students to join the points or dots that form the figure of a person, object cute animal or form a trail or path in the map with some amusing significance - path out of a maze, path between two cities following a road network,  path to buried treasure, etc, etc - where the etc, etc means I have run out of imagination. The introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions.  
   
5. Map Usage:  From measurement and scaling of map coordinates, students may find the physical location of a point, or its image on another map.  Maps may also be used to draw and plan routes.  From measurement and scaling of map lengths with rulers, threads and measuring wheels (official name?), students may obtain the physical length of routes.  Bearing (angles) of a distance object and the endpoints of the line segment joining two bearings would allow students to locate on the map the distant object using the ASA method. The foregoing may be combined with more map mastery exercises.

8 More on Geometry with Maps, Plans and Designs

Odds and Ends:

1. Arrows and Navigation: Actual or potential path (trips, voyages, routes) may shown on maps by curves - smooth or piecewise linear. The net result of a trip is a movement or displacement from the initial point (origin of the path) to the terminal point that can represented (drawn) as an arrow or vector with tail at the initial point and head at the terminal point. Paths that involve a sequence of net movements from one point to a next can be represent by piecewise linear curves in which linear part, an actual or net linear displacement,  is represented by an arrow.  The observation of a first displacement and then a second displacement and then a third displacement, etc, leads to the adding of displacements or arrows or vectors in a head to tail manner.  Diagrams imply that addition is clearly associative. 
   
   Technical Remark: The Head to Tail Addition of a sequence of displacements (arrows), where subsequences are replaced by a net result (resultant vector) is  associative. [insert picture to demonstrate] Moreover, adjacent element of the sequence (subsequences) can be grouped and replaced by a single resultant arrow, all without changing the net displacement from the initial point of the route to the terminal point of the route. 
   
2. Orienteering:  Walking through parks and bush with the aid of maps and compass may introduce veering-off strategy of aiming for one side of a desired destination instead of heading directly for it.  For example if the destination is on a stream and there is some uncertainty or impossibility of heading directly for that destination, the orienteer would set a course or direction that guarantees hitting the stream above (or below) the destination, and then plan to walk downstream (respectively upstream) to the desired destination.  The alternative might lead the orienteer to the stream without being certain of being upstream or downstream of the target location.
   
   
3. Plans and Maps in Design and Navigation:  Students may be shown how to solve geometric design and navigation problems by drawing objects and paths to scale and then measuring lengths on the drawing to compute actual lengths and from them compute areas, and other quantities proportional to length and area. Here is a context for the introduction and study of similarity since  corresponding Angles are preserved and corresponding lengths are proportional in the drawing,  design and use of shapes and routes on maps and plans.  The study of similarity may focus on the use of scale drawing to draw conclusion about real or imagine world situations and applications with the aid of geometric formulas and real or on-map measurements. Applications may appear in interior design - the painting and design of rooms in homes and offices, and the calculation of quantities based on length, area and even volume. Application may also appear in surveying - determining heights of building from angle measurements and horizontal distances.  Application may also appear in navigation - route planning for vehicles on land, on water, or over and under land and water.  The common theme may drawing to scale on map or plans, and then rescaling map measurements to get or estimate real world measures. 
   
4. Navigation and Movements with arrows and vectors on Maps.    The planning of routes at sea and in the air may involve straight line segments with an initial and terminal point.  Each segment may be depicted by an arrow or vector with a tail starting at the initial point and a head ending at the terminal point. A piecewise linear route (top view) in the plane may be represented by a sequence of arrows, with the tail of the first arrow at the initial point of the route, and the tail at each further arrow at the head of is predecessor.  Each arrow represents a displacement from its initial point or tail to its terminal point or head. The arrow points in the direction of movement. So an arrow in depicting a displacement has a length (magnitude) and direction. The arrow also has an initial position (the tip of its tail) and a terminal position - the tip of its head. The head to tail placement of arrows represent a pair or sequence of displacements, a route in which the end of one displacement is the start of the next. The net displacement of a such a sequence is the arrow from the initial point of the route to the terminal point.  That arrow represents the net displacement of two to several displacements. In particular, the net result (sum or resultant) of two adjacent arrows or displacements in the route may be represented by a third arrow or displacement that starts at the initial point of the first arrow and ends at the terminal point of the second arrow. 
   
   A sequence of movements (displacements) plotted as arrows may give or approximate an actual or planned route. The map location of heads and tails may correspond to points on the route where bearing were taken to determine location on the map, or those map location may represent the intended location.  Maps and charts may show the intended and actual route of a vessel or vehicle across the sea or land.

Remark on Parallelism:   On maps and plans, the practice of indicating true North or Magnetic North provides a reference for describing the direction of line segments and movement. Two movement are declared to be parallel if they make the same bearing or angle with respect to true North or magnetic North.  Parallel movements are thus indicated by compass readings. Implicit here is theorem that could be stated and shown in particular cases, if not in general,  in Euclidean Geometry:  Namely, if at two distinct  points are located on parallel lines, and rays issuing from the points make congruent angles with the parallel lines, then the two rays and lines extending them are parallel. Likewise, the practice of covering maps and plans with grid also provides a reference for describing the direction of line segments and movement. How these practices are possible might be a subject of discussion in Euclidean Geometry - the applied math view of it.