Ex. 372. Find the locus of a point at a given distance from a given straight line.

Ex. 373. Find a point which is equidistant from three given points not in the same straight line.

Ex. 374. Find the locus of a point equidistant from the circumferences of two concentric circles.

Ex. 375. Find a point in a given straight line which is equidistant from two given points.

Ex. 376. Find the locus of the center of a circle tangent to each of two parallel lines.

Ex. 377. Find the locus of the center of a circle which touches a given line at a given point.

Ex. 378. Find the locus of the center of a circle of given radius that passes through a given point.

Ex. 379. Find the locus of the center of a circle which is tangent to a given circle at a given point.

Ex. 380. Find the locus of the center of a circle of given radius and tangent to a given circle.

Ex. 381. Find the locus of the center of a circle passing through two given points.

Ex. 382. Find the locus of the center of a circle of given radius and tangent to a given line.

Ex. 383. Find the locus of the center of a circle tangent to each of two intersecting lines.

Ex. 384. Find the locus of the middle points of a system of parallel chords drawn in a circle.

Ex. 385. Find the locus of the middle points of equal chords of a given circle.

Ex. 386. Find the locus of the extremities of tangents of fixed length drawn to a given circle.

Ex. 387. Find the locus of the middle point of a straight line drawn from a given point to meet a given straight line.

Ex. 388. Find the locus of the vertex of a right triangle on a given base as hypotenuse.

Ex. 389. Find the locus of the middle points of all chords of a circle drawn from a fixed point in the circumference.

Ex. 390. Find the locus of the middle point of a straight line moving between the sides of a right angle.

Ex. 391. Find the locus of the points of contact of tangents from a fixed point to a system of concentric circles.

Ex. 392. Find the locus of the middle points of secants drawn from a given point to a given circle.

BOOK III

RATIO AND PROPORTION

257. 1. How is a magnitude measured?

2. What is the numerical measure of a magnitude?

3. What is the common measure of two or more magnitudes ? 4. What is meant by the ratio of two magnitudes?

5. How may the ratio of two magnitudes be determined?

6. Since the ratio of two magnitudes is the ratio of their numerical measures, what is the relation of two magnitudes whose numerical measures are 8 and 16 respectively? 5 and 10? 12 and 36? 15 and 45?

7. How does 8 compare with 2?

What is the relation of 3

to 9? Of 12 to 4? Of 18 to 3? Of 20 to 40? Of 25 to 75? Of 35 to 70 ?

8. What is the ratio of 1 ft. to 1 yd.? 3 in. to 1 ft.? 2cm to 1dm? 5dm to 2m? 2 sq. ft. to 2 sq. yd.? 3 cu. ft. to 1 cu. yd.?

258. The quantities compared are called the Terms of the ratio. A ratio is denoted by a colon placed between the terms.

The ratio between 2 and 5 is expressed 2: 5.

259. The first term of a ratio is called the Antecedent of the ratio. The second term of a ratio is called the Consequent of the ratio.

260. The antecedent and consequent together form a Couplet.

261. Since the ratio of two quantities may be expressed by a fraction, as , it follows that:

α

The changes which may be made upon the terms of a fraction without altering its value may be made upon the terms of a ratio without altering the ratio.

262. 1. What two numbers have the same relation to each other as 3 has to 6? 2 to 8? 5 to 15? 8 to 4?

2. What numbers have the same relation to each other that 4 in. has to 2 ft.? 5 ft. to 2 yd.? 5cm to 1m? 3dm to 8cm ?

3. What number has the same relation to 6 that 2 has to 4? 4. What number has the same relation to 12 that 3 has to 9? 5. What number has the same ratio to 8 that 5 has to 15?

263. An equality of ratios is called a Proportion.

The sign of equality is written between the equal ratios.

a b c d is a proportion, and is read: the ratio of a to b is equal to the ratio of c to d, or a is to b as c is to d.

The double colon,::, is. frequently used instead of the sign of equality.

264. The antecedents of the ratios which form a proportion are called the Antecedents of the proportion, and the consequents of those ratios are called the Consequents of the proportion.

In the proportion a: bc: d, a and c are the antecedents, and b and d are the consequents of the proportion.

265. The first and fourth terms of a proportion are called the Extremes and the second and third terms are called the Means of the proportion.

In the proportion a: bc: d, a and d are the extremes, and b and c are the means.

266. A quantity which serves as both means of a proportion is called a Mean Proportional.

In the proportion a: b = b : c, b is a mean proportional.

267. Since a proportion is an equality of ratios, and the ratio of one quantity to another is found by dividing the antecedent by the consequent, it follows that:

A proportion may be expressed as an equation in which both members are fractions.

The proportion a: b = c. d may be written

a с

b

d

Such an expression is to be read as the ordinary form of a proportion is

read.

268. Since a proportion may be regarded as an equation in which both members are fractions, it follows that:

1. The changes which may be made upon the members of an equation without destroying its equality may be made upon the couplets of a proportion without destroying the equality of the ratios.

2. The changes which may be made upon the terms of a fraction without altering the value of the fraction may be made upon the terms of each ratio of the proportion without destroying the proportion.

Proposition I

269. 1. Form several proportions, as 3:59:15, and discover how the product of the extremes compares with the product of the means in each.

2. If the means in any proportion are the same, how may the means be found from the product of the extremes?

3. Form a proportion whose consequents are equal. How do the antecedents compare?

4. Form a proportion in which either antecedent is equal to its consequent. How does the other antecedent compare with its consequent?

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

270. Cor. I. A mean proportional between two quantities is equal to the square root of their product.

If a: b = b: c, find the value of b.

271. Cor. II. If in any proportion any antecedent is equal to its consequent, the other antecedent is equal to its consequent.

272. Cor. III. If the consequents of any proportion are equal, the antecedents are equal, and conversely.

273. 1. If the product of the extremes of a proportion is 48, what may the extremes be? If 72? If 30? If 36? If 6 a2? If 12 ab? If abc? 2. If the product of the means is 48, what may the means be? If 96? If 108? If 6 bed? If abcd? If a2b2? If abc?

3. Form a proportion the product of whose extremes or means is 60; 72; 84; 80; 64; 144; x2y2, xyz; xyzv.

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

Ex. 393. If the vertical angle of an isosceles triangle is 30°, what is its ratio to each of the base angles?

Ex. 394. If the exterior angle at the base of an isosceles triangle is 100°, what is its ratio to each angle of the triangle ?

Ex. 395. If one of the acute angles of a right triangle is 40°, what is its ratio to the other acute angle? To the right angle?

Ex. 396. The interior angles on the same side of a transversal cutting two parallel lines are to each other as 3 to 2. How many degrees are there in each angle?

Ex. 397. The vertical angle of an isosceles triangle has the same ratio to a right angle that an angle of 40° has to an angle of an equilateral triangle. How many degrees are there in each angle of the isosceles triangle?