CHAPTER XVII

SIMILAR TRIANGLES

The basic method is:

*257. Two triangles are similar if two angles of the one are equal respectively to two angles of the other.

*258. Two triangles are similar if an angle of the one equals an angle of the other and the including sides are in proportion.

259. What can be said of the corresponding sides of similar triangles? The corresponding angles?

260. What are the basic theorems of similar triangles? How are the other theorems proved?

261. Two right triangles are similar if an acute angle of the one equals an acute angle of the other.

262. Two right triangles are similar if the two arms of the one are in proportion with the two arms of the other.

263. Two right triangles are similar if the hypotenuse and arm of the one are in proportion with the hypotenuse and arm of the other.

264. Two triangles are similar if their corresponding sides are in proportion.

265. From Articles 261, 262, and 263, a general theorem for all right triangles can be made:

Two right triangles are similar if an acute angle of the one equals an acute angle of the other, or if two sides of the

one are in proportion with two corresponding sides of the other.

*266. PROBLEM: On a given straight line construct a triangle similar to a given triangle.

SUMMARY

267. State four methods of proving triangles similar, covering all cases.

What is the principal use of similar triangles? How can the corresponding sides be found?

268.

1.

EXERCISES

If two triangles are mutually equiangular, they are similar. 2. The line crossing two sides of a triangle and parallel to the third side cuts off a triangle similar to the given triangle.

3. Two isosceles triangles are similar if an angle of the one equals a corresponding angle of the other.

4. If E and H are points on side AB of ZABC, and EF and HK are 1 BC, A EBF ▲ HB K.

5. If two chords AB and CD intersect at E within a circle, ▲AEC ABED.

6. If two chords AB and CD intersect at E without a circle, when extended, ▲ AED ~ ▲ BEC.

7. Two triangles are similar if their sides are respectively parallel each to each.

8. Two triangles are similar if their sides are respectively perpendicular each to each.

9. Two circles are tangent externally at H, and through H three lines are drawn meeting one circle in A, B, and C, and the other circle in D, E, and F. Prove ▲ ABC ~ ▲ DEF.

10. If BD and CE are altitudes of Δ ABC, Δ ABD - Δ ACE.

11. In the same figure, if BD and CE intersect at O, ABOE ▲ COD. In the same figure, prove ▲ ABC ~ ▲ AED.

12.

13. The lines joining the middle points of the sides of a triangle form a triangle similar to the given triangle.

CHAPTER XVIII

PROPORTION FROM SIMILAR TRIANGLES

The basic method is:

The corresponding sides of similar triangles are in proportion. (Art. 259.)

*269. The corresponding altitudes of two similar triangles are in proportion with any two corresponding sides.

270. State the method of proving that the product of two quantities equals the product of two other quantities. *271. If two chords intersect within a circle, the product of the parts of one chord equals the product of the parts of the other chord.

*272. If two secants intersect without a circle, the product of one whole secant and its external part equals the product of the other whole secant and its external part.

273. If a secant and a tangent intersect without a circle, the product of the whole secant and its external part equals the square of the tangent.

274. Group 271, 272, and 273 under one statement. 275. Define Mean Proportional. Third Proportional. *276. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse:

1. The triangles formed are similar to the given triangle and to each other.

2. The perpendicular is the mean proportional between the parts of the hypotenuse.

3. Either arm is the mean proportional between the whole hypotenuse and its adjacent part.

277. The perpendicular from any point in the circumference of a circle to the diameter is a mean proportional between the parts of the diameter.

*278. PROBLEM: Construct a mean proportional between two given straight lines.

SUMMARY

279. State two new methods of proving lines in proportion. Recall the other two.

280. State three methods of proving a line a mean proportional between two other lines.

1. The corresponding medians of two similar triangles are in proportion with any two corresponding sides.

2. The corresponding angle-bisectors of two similar triangles are in proportion with any two corresponding sides.

3. The diagonals of a trapezoid divide each other proportionally. 4. The radii of the circles inscribed in two similar triangles are in proportion with any two corresponding sides.

5. The radii of the circles circumscribed about two similar triangles are in proportion with any two corresponding sides.

6. In any right triangle, the product of the hypotenuse and the altitude upon it equals the product of the arms.

7. The product of any altitude of a triangle and its corresponding side equals the product of any other altitude and its corresponding side.

8. Find a mean proportional between two lines a and b by three different methods.

9. If a and c are the extremes of a proportion, show that the mean proportional x = √ ac.

10. Find a mean proportional between 1 and 4, arithmetically and geometrically.

b2

11. If x is a third proportional to a and b, show that x=—.

α

12. Find a third proportional between lines a and b. 13. Find a third proportional to 2 and 3, arithmetically and geometrically.

14. Find geometrically √3; √2; √5.

CHAPTER XIX

THE PYTHAGOREAN THEOREM

*282. The square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.

SUMMARY

283. State the formula for the hypotenuse c in terms of a and b.

State the formula for a in terms of c, if ≤ A =60°.

State the formula for c in terms of b, if A = 45°.

State the four steps in the analysis of numerical exercises.

284.

NUMERICAL EXERCISES

1. How long a ladder is required to reach a window 24 ft. high, if the lower end of the ladder is 10 ft. from the side of the house?

2. Find the altitude of an equilateral triangle whose side is 1 in. 3. Find the sides of a right triangle whose hypotenuse is 1 in., one of the acute angles being 60°.

4. Find the hypotenuse of a right triangle whose side is 1 in., one of the acute angles being 45°.

5. Find the length of the shortest chord and of the longest chord that can be drawn through a point 6 in. from the center of a circle whose radius is 10 in.

6. The distance from the center of a circle to a chord 10 ft. long is 12 ft. Find the distance from the center to a chord 24 ft. long.

7. The radius of a circle is 6 in. Find the length of the tangents drawn from a point 10 in. from the center, and also the length of the chord between the points of tangency.

8. The radius of a circle is 2 in. From a point 4 in. from the center, a secant is drawn so that its internal part is 1 in. Find the length of the secant.

9. If one angle of a right triangle is 60°, and the opposite side is 2 √3, find the other sides. (See Art. 120.)

10. If one angle of a right triangle is 45°, and the hypotenuse is 5 in., find the other sides.

11. A window 5 ft. wide has a circular arch at the top with a rise of 20 in. Find the radius of the circle of the arch.