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12. The non-parallel sides of a trapezoid are each 53 units in length, and one of the bases is 56 units longer than the other. Find the altitude of the trapezoid.

NUMERICAL EXERCISES ON PROPORTION 1. The base and altitude of a triangle are 7 ft. 6 in. and 5 ft. 6 in. respectively. If the corresponding base of a similar triangle is 5 ft. 6 in., find the altitude.

2. The sides of a triangle are 6,7, and 8. In a similar triangle, the side corresponding to 8 is 40. Find the other two sides.

3. The sides of a right triangle are 9, 12, and 15. Find the parts of the hypotenuse made by the altitude. Find this altitude.

4. The radius of a circle is 13 in. Through a point 5 in. from the center, a chord is drawn. What is the product of the two parts of the chord?

5. From the end of a tangent 20 in. long, a secant is drawn through the center of the circle. If the external part of the secant is 8 in., find the radius of the circle.

6. A tree casts a shadow 90 ft. long, when a post 6 ft. high casts a shadow 4 ft. long. How high is the tree?

7. Construct ex. 13, Art. 86, with similar triangles instead of congruent triangles.

8. Construct ex. 14, Art. 86, similarly. 9. Construct ex. 15, Art. 86, similarly. 10. When the sun shines upon the earth, the earth casts a shadow in space. This shadow is represented on paper by a triangle. Thus:

B

s

In the figure, S is the center of the sun; E is the earth; EW is the length of the shadow. Find EW. Given: ES = 92,000,000 miles; SD = 433,000 miles; EB = 4,000 miles.

CHAPTER XX

SIMILAR POLYGONS AND CIRCLES

The basic method is:

Similar polygons have their corresponding angles equal and their corresponding sides in proportion. (Art. 253.)

*285. Two polygons are similar, if they are composed of the same number of triangles, similar each to each, and similarly placed.

*286. Two similar polygons may be divided into the same number of triangles, similar each to each, and similarly placed.

*287. Problem: On a given line corresponding to a side of a given polygon, construct a polygon similar to the given polygon.

288. Two regular polygons of the same number of sides are similar.

289. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as .. what?

*290. The perimeters of two similar polygons are to each other as any two corresponding sides.

*291. The perimeters of two regular polygons of the same number of sides are to each other as their radii and as their apothems.

292. What is a Constant, a Variable, a Limit? What is the usual variable considered in geometry?

293. If two variables are always equal, and if each approaches a limit, their limits are equal.

*294. The circumferences of two circles are to each other as their radii.

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295. The ratio of a circumference to its diameter equals the ratio of any other circumference to its diameter.

296. What does a represent? What is the formula for the circumference (C) in terms of . ?

297. Problem: Show that a > 3, and < 4. Its commonly accepted value is what?

SUMMARY

298. State two methods of proving polygons similar.

299.

NUMERICAL EXERCISES

1. The perimeters of two similar polygons are 200 ft. and 300 ft. If a side of the first is 24 ft., find the corresponding side of the second.

2. Find the circumference of a circle whose diameter is 8 ft.

3. The diameter of a bicycle wheel is 28 in. How many revolutions does the wheel make in going one mile?

4. Find the diameter of a carriage wheel that makes 264 revolutions in going half a mile.

5. Find the central angle of an arc 5 ft. 10 in. long, if the radius is 9 ft. 4 in.

6. Find the arc whose central angle is 40°, if the radius is 6 in.

7. Find the width of a circular ring between two concentric circles whose circumferences are 130 ft. and 85 ft. respectively.

8. A water-tank is cylindrical in form and 20 ft. in diameter. How long must a piece of strap iron be cut to make a band around it, allowing 20 in. for overlapping?

9. The earth is nearly 8,000 miles in diameter. What is the approximate length of the equator?

10. Two wheels are each 24 in. in diameter and their centers are 8 ft. apart. What is the length of the belt that runs around the two wheels?

11. A driving wheel 4 ft. in diameter makes 100 revolutions per minute. It is belted to a second wheel that makes 50 revolutions per minute. What is the diameter of the second wheel?

12. The side of a square is 4 ft. Find the circumference of its inscribed and circumscribed circles.

REVIEW QUESTIONS

1. Define Ratio; Proportion; Extremes and Means; Antecedents and Consequents; Alternation; Inversion; Composition, and Division in proportion.

2. What is meant by: The product of the means equals the product of the extremes? What is its converse?

3. A proportion may be considered as a fractional what?

4. If three terms of one proportion are equal respectively to three terms of another proportion, what can be said of the remaining terms?

5. What is the fundamental principle for all proportion of lines? 6. What is a more common form of this principle? 7. Define Internal division of a line; External division of a line? 8. Define Similar polygons. 9. State four methods of proving triangles similar. 10. Which is shorter, proving triangles similar or other polygons? 11. How can the corresponding sides of similar triangles be found? 12. State four methods of proving four lines in proportion.

18. There are now nine methods of proving lines parallel. Name most of them.

14. There are now seventeen methods of proving angles equal. Name as many of them as possible.

15. What is the usual method of getting four lines in proportion? 16. The ratio of two lines means the ratio of what?

17. State the three steps in proving that the product of two lines equals the product of two other lines.

18. Group 271, 272, and 273 under one statement.

19. Define Mean proportional; Third proportional; Fourth proportional.

20. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, state three conclusions.

21. State three methods of finding a mean proportional to two given straight lines.

22. Represent algebraically the fourth proportional to a, b, and c. 23. Represent algebraically the mean proportional between a and b. 24. What is the Pythagorean theorem? 25. What is its formula? 26. State two important methods of proving polygons similar. 27. What conclusion can be made from a series of equal ratios? 28. What is a Constant? Variable? Limit? 29. What is the usual variable considered in geometry?

MISCELLANEOUS EXERCISES

1. If two transversals intersect between two parallel lines, the triangles formed are similar.

2. The diagonals of a trapezoid form with the bases two similar triangles.

3. If two circles are tangent internally and three lines are drawn through the point of tangency, the chords joining the ends of these lines form similar triangles.

4. If two circles are tangent externally and two lines are drawn through the point of tangency, the chords joining the ends of these lines form two similar triangles.

5. Prove the same theorem if the circles are tangent internally.

6. If triangle ABC is inscribed in a circle and AP is drawn to the middle point of arc BC, and cutting chord BC in D; triangle ABD is similar to triangle A PC.

7. If triangle ABC is inscribed in a circle, and AD is drawn I to BC, and the diameter AE is drawn; triangle ABD is similar to triangle AEC.

8. The median to the base of a triangle bisects any line cutting the triangle and parallel to the base.

9. The squares of the arms of a right triangle are to each other as their projections on the hypotenuse.

10. Two rhombuses are similar if an angle of the one equals an angle of the other.

11. Two parallelograms are similar if an angle of the one equals an angle of the other and the including sides are proportional.

12. Two rectangles are similar, if two adjacent sides of the one are in proportion with two corresponding sides of the other.

13. An interior common tangent of two circles divides the line of centers proportionally with the radii.

14. An exterior common tangent of two circles divides the line of centers externally into parts proportional with their radii.

15. If three or more straight lines drawn from a common point intersect two parallels, the corresponding parts of the parallels are proportional.

16. If two circles are tangent externally, their common external tangent is a mean proportional between their diameters.

17. If two circles are tangent internally, all chords of the greater circle drawn from the point of tangency are divided proportionally by the circumference of the smaller circle.

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