Mth 075: Applied Geometry (Individualized Sections) MODULE THREE STUDY GUIDE INTRODUCTION TO GEOMETRY Assignment Seven: Problems Involving Right Triangles A. Read pages 35-38 in your textbook. Study examples 1-7 odd on pages 38 & 39. B. Work examples 2-8 even on pages 38 & 39 and check your results. C. Assignment Seven: Do problems 2-10 even, 14, 1, 20-2 even, 30 and 32 in Exercise 3, Objective A, pages 45 & 4. Assignment Eight: Problems Involving Similar Triangles A. Read pages 40 & 41 in your textbook. Study examples 9, 11 on pages 41 & 42. B. Work examples 10, 12 on pages 41 & 42 and check your results. C. Assignment Eight: Do problems 34, 38-48 even, 52, 54, 5 in Exercise 3, Objective B, pages 47, 48, 49. Assignment Nine: Problems Involving Similar Polygons and Scale Drawings A. Read and study pages 2-4 in the study guide. B. Assignment Nine: Do problems 1-10 on pages 5 & of this study guide. Assignment Ten: Problems Involving Congruent Triangles A. Read pages 42 & 43 in your textbook. Study examples13 & 15 on page 44. B. Work examples 14 & 1 on page 44 and check your results. C. Assignment Ten: Do problems 58-8 even in exercise 3, objective C, pages 49 & 50. Review Assignment: Do the module three review assignment available in the math learning center and submit it to an instructor for grading. Module Test: After successfully completing the four assignments and the review assignment, ask an instructor for a module three test. 1
Mth 075 Module 3 Supplement Similar Polygons At this point, we have looked at the two properties of similar triangles; the corresponding angles of similar triangles are equal in measure and the corresponding lengths of sides of similar triangles are proportional. Now let s look at the situation in reverse to see what properties are necessary to show that two triangles are similar. If the measures of two angles of one triangle are equal to the measures of two angles of another triangle, the triangles are similar. Notice that if this requirement is met, the third angle in each triangle must be equal in measure since each triangle s interior angles total 180 o. This is one way to show that two triangles are similar. B E 5 o 37 o A 37 o 5 o F C D ABC is similar to DEF. A second way to show that two triangles are similar is to show the lengths of corresponding sides are proportional. T Q 8 12 1 V 5 U R 10 S QRS is similar to TUV since side RS corresponds to side UV, side TU corresponds to side QR, side QS corresponds to side TV and 5 10 = 12 = 8 1. Notice we are saying that only one of the two situations must be met to show that two triangles are similar. Let s look at a couple of examples of other polygons that are not triangles to see if this continues to be true. 2
These two figures show that a polygon that is not a triangle can have corresponding angles equal in measure and the polygons are not similar. These two polygons are clearly not similar even though all of the corresponding angles are equal to 90 o. 3 2 These two figures show that a polygon that is not a triangle can have corresponding sides 2 2 3 3 proportional and the polygons are not similar. 2 3 These figures are clearly not similar even though their corresponding sides all have a ratio of 2 3. Therefore, if two polygons that are not triangles are to be shown to be similar, one must show that corresponding angles are equal and the lengths of corresponding sides are proportional. Scale Drawings One of the most practical uses of similar geometric figures is that of scale drawings. Maps, charts and blueprints are familiar examples of scale drawings. In a scale drawing, all distances are drawn a certain ratio of the distances they represent. Since a scale drawing is similar to the original object, the ratios of corresponding lengths are equal. This makes it possible to indirectly measure inaccessible or large distances or to create drawings that allow one to determine actual lengths. The drawing on the next page is a scale drawing of measurements taken by a ranger in order to approximate the distance across a lake. Note the given scale of 1 inch = 100 feet. Since the drawing has been made to scale, the ratios of the length of a segment in the drawing in inches to 1 inch must equal the actual length in feet to 100 feet. 3
Scale: 1 inch = 100 feet C 9.4 o 240 ft. 325 ft. A B To determine the length across the lake, measure the length of the segment AB in inches and set up the corresponding proportion. AB measures 4.25 inches in the scale drawing. 425. " ' = AB. Solving this equation for AB gives AB = 425 feet. 1" 100' The scale drawing below represents the foundation for a house. Use the scale drawing to determine the actual length and width of the foundation. Scale: 1 in. = 10 ft. Measuring the length and width with a ruler, we find that the length of the scale drawing is 3 inches and the width of the scale drawing is 2 inches. Using the given scale and these measurements, we can now set up the appropriate proportions. To find the actual length we use the proportion: 3 " = L. Solving this equation for L gives 1" 10' L = 30 feet. To find the actual width we use the proportion : 2 " = W. Solving this equation for W gives 1" 10' W = 20 feet. 4
Assignment # 9 Determine which of the following polygons are similar. 1. 27 o 3 o 2. 2 4 4 8 10 3. 10 20 5 10 4. 135 o 135 o 5 10 5. 3 Parallelograms 9 3. 2 0 o Parallelograms 0 o 5
7. 3 5 70 o Rhombi 70 o A E Consider ABC and EDC. AB ED 8. B D C Scale: 1 = 10 beam 9.? The scale drawing above represents the floor design for a cabin. A beam is needed to span an opening to one of the rooms. Using the given scale, determine the length of the beam needed to span this opening. 10. Make your own scale drawing of the information in the sketch to determine the height of the mountain. Note: This drawing is not drawn to scale.? 45 o 0 o 1000 ft.