Ex. 353. Construct a circle to pass through a given point and be tangent to a given circle at a given point. When is this impossible?

Ex. 354. Construct a circle to pass through a given point and touch a given straight line at a given point.

Ex. 355. Construct a circle to touch three given straight lines.

Ex. 356. Within an equilateral triangle, to describe three circles each tangent to the other two and to two sides of the triangle.

Ex. 357. Construct a circle of given radius to touch two given straight lines.

Ex. 358. Construct a circle of given radius, having its center on a given straight line and touching another given straight line. How many solutions may there be ?

Ex. 359. Construct a right triangle, having given the radius of the inscribed circle and one of the sides containing the right angle.

Ex. 360. Construct a triangle, having given the base, the vertical angle, and the length of the median to the base.

Ex. 361. Construct a triangle, having given the three middle points of its sides.

Ex. 362. Construct a circle of given radius to pass through a given point and touch a given straight line.

Ex. 363. From the vertices of a triangle as centers, to describe three circles which shall be tangent to each other.

Ex. 364. Construct a triangle, having given the base, altitude, and radius of the circumscribed circle.

Ex. 365. Three given straight lines meet at a point; draw another straight line so that the two portions of it intercepted between the given lines are equal. How many solutions may there be?

SUGGESTION. Form a parallelogram.

Ex. 366. Through a given point, between two intersecting straight lines, to draw a line terminated by the given lines and bisected at the given point. Ex. 367. Construct a circle to intercept equal chords of given length on three given straight lines.

Ex. 368. Construct a triangle, having given one angle, the opposite side, and the sum of the other two sides.

THE LOCUS OF A POINT

256. When a point equidistant from the extremities of a straight line is to be found, the middle point of the line meets the conditions. But there are other points which also fulfill the required conditions, for every point in the perpendicular to the given line at its middle point is equidistant from the extremities of the line.

Such a perpendicular is called the locus of the points which are equidistant from the extremities of the line.

The line, or system of lines, containing every point which satisfies certain given conditions, and no other points, is called the Locus of those points.

A locus may also be described as a line, or the lines, traced by a point which moves in accordance with given conditions.

To prove that a certain line, or system of lines, is the required locus, it must be shown:

1. That every point in the lines satisfies the given conditions.

2. That any point not in the lines cannot satisfy the given conditions.

Ex. 369. Find the locus of a point which is equidistant from two intersecting straight lines.

Data: Any two straight lines, as AB and CD, intersecting at the point K.

Required to find the locus of a point equidistant from AB and CD.

Solution.

E

B

A point equidistant from two inter

M

G

-H

secting straight lines suggests a point in the bisector of an angle.

A

Draw EF bisecting & CKB and AKD, and also GH bisecting AKC and BKD.

Then, § 134, every point in EF is equidistant from AB and CD, and

every point in GH is equidistant from AB and CD.

If all other points are unequally distant from AB and CD, then EF and GH is the required locus.

From Pany point without the bisectors draw PM 1 CD, and PR 1 AB, intersecting EF in J.

From J draw JL 1 CD, and also draw PL.

That is, the point P is unequally distant from AB and CD.

Hence EF and GH is the required locus.

Why?

Ex. 370. Find the locus of a point at a given distance from a given point.

Ex. 371. Find the locus of a point equidistant from two parallel straight lines.

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 points of straight lines 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? 5am 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:

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: b = c: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: b = c: 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: bc: d may be written

α с

=

b

d

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

read.