27. Construct this design, making the figure twice this size.

Construct the equilateral ▲. Then describe the small with half the side of the A as a radius. Then find the radius of the circumscribing O.

28. A circular window in a church has a design similar to the accompanying figure. Draw it, making the figure twice this size.

This is made from the figure of the preceding exercise, by erasing certain lines.

29. Two wheels of radii 1 ft. 6 in. and 2 ft. 3 in. respectively are connected by a belt, drawn straight between the points of tangency. The centers being 6 ft. apart, draw the figure mathematically. Use the scale of 1 in. to the foot.

30. A water wheel is broken and all but a fragment is lost. A workman wishes to restore the wheel. Make a drawing showing how he can construct a wheel the size of the original.

31. In this figure ▲m=62°, and ≤n= 28°. Find the number of degrees in each of the other angles, and determine whether AB is a diameter.

32. In this figure B=41°, ZA=65°, and BDC= 97°. Find the number of degrees in each of the other angles, and determine whether CD is a diameter.

A

B

m

B

33. Construct or explain why it is impossible to construct a triangle with sides 3 in., 2 in., 6 in.; also one with sides 5 in., 7 in., 12 in.; also one with sides 2 in., 1 in., 11⁄2 in.

34. Show how to draw a tangent to this circle at the point P, the center of the circle not being accessible.

P

EXERCISE 40

1. In a circle whose center is O the chord AB is that BA0=27°. How many degrees are there in

drawn so

AOB?

2. In a circle whose center is O the chord AB is drawn so that BAO 25°. On the circle, and on the same side of AB as the center O, the point D is taken and is joined to A and B. How many degrees are there in ZADB?

3. What is the locus of the mid-point of a chord of a circle formed by secants drawn from a given external point?

4. In a circle whose center is O two perpendiculars OM and ON are drawn to the chords AB and CD respectively, and it is known that ▲ NMO = LONM. Prove that AB = CD.

5. Two circles intersect at the points A and B. Through A a variable secant is drawn, cutting the circles at C and D. Prove that the angle DBC is constant.

6. Let A and B be two fixed points on a given circle, and M and N be the extremities of a rotating diameter of the same circle. Find the locus of the point of intersection of the lines AM and BN.

7. Upon a line AB a segment of a circle containing 240° is constructed, and in the segment any chord PQ subtending an arc of 60° is drawn. Find the locus of the point of intersection of AP and BQ; also of AQ and BP.

8. To construct a square, given the sum of the diagonal and one side.

Let ABCD be the square required, and CA the di- E agonal. Produce CA, making AE= AB. §ABC and ABE are isosceles and LBAC = LACB= 45°. Find

the value of E. Construct 2CBE. Now reverse the reasoning.

The propositions in Exercise 40 are taken from recent college entrance examination papers.

EXERCISE 41

REVIEW QUESTIONS

1. Define the word circle and the principal terms used in connection with it.

2. What is meant by a central angle? How is it measured? 3. What is meant by an inscribed angle? How is it measured? 4. State the general proposition covering all the cases that have been considered relating to the measure of an angle formed by the intersection of two secants.

5. State all of the facts you have learned relating to equal chords of a circle.

6. State all of the facts you have learned relating to unequal chords of a circle.

7. State all of the facts you have learned relating to tangents to a circle.

8. How many points are required to determine a straight line? two parallel lines? an angle? a circle?

9. Name one kind of magnitude that you have learned to trisect, and state how you proceed to trisect this magnitude. 10. In order to construct a definite triangle, what parts must be known?

11. What are the important methods of attacking a new problem in geometry? Which is the best method to try first?

12. What is meant by determinate, indeterminate, and impossible cases in the solution of a problem?

13. Distinguish between a constant and a variable, and give an illustration of each.

14. Distinguish between inscribed, circumscribed, and escribed circles.

15. What is meant by the statement that a central angle is measured by the intercepted arc?

BOOK III

PROPORTION. SIMILAR POLYGONS

256. Proportion. An expression of equality between two equal ratios is called a proportion.

257. Symbols. A proportion is written in one of the following forms:

α с

b

=; a:b=c:d; a:b::c:d.
;;
d

This proportion is read "a is to b as c is to d"; or" the ratio of a to b is equal to the ratio of c to d."

258. Terms. In a proportion the four quantities compared are called the terms. The first and third terms are called the antecedents; the second and fourth terms, the consequents. The first and fourth terms are called the extremes; the second and third terms, the means.

Thus in the proportion a:bc:d, a and c are the antecedents, b and d the consequents, a and d the extremes, b and c the means.

259. Fourth Proportional. The fourth term of a proportion is called the fourth proportional to the terms taken in order. Thus in the proportion a:bc:d, d is the fourth proportional to a, b, and c.

...

are

260. Continued Proportion. The quantities a, b, c, d, said to be in continued proportion, if a:b=b:cc: d If three quantities are in continued proportion, the second is called the mean proportional between the other two, and the third is called the third proportional to the other two.

Thus in the proportion a:b

b:c, b is the mean proportional between

a and c, and c is the third proportional to a and b.

PROPOSITION I. THEOREM

261. In any proportion the product of the extremes is equal to the product of the means.

Given

To prove that

Proof.

Multiplying by bd,

262. COROLLARY 1.

The mean proportional between two quantities is equal to the square root of their product. For if a:bb: c, then b2 = ac (§ 261), and b = √ac, by Ax. 5. 263. COROLLARY 2. If the two antecedents of a proportion are equal, the two consequents are equal.

264. COROLLARY 3. If the product of two quantities is equal to the product of two others, either two may be made the extremes of a proportion in which the other two are made the means.

For if ad = bc, then, by dividing by bd,=, by Ax. 4.

PROPOSITION II. THEOREM

265. If four quantities are in proportion, they are in proportion by alternation; that is, the first term is to the third as the second term is to the fourth.