g. If their sides are perpendicular, each to each.

§ 307

h. If they are the corresponding triangles of similar polygons divided by homologous diagonals.

§ 310

7. Two polygons are similar,

a. If they have their homologous angles equal and their homologous sides proportional.

$ 299

b. If each is composed of the same number of triangles similar each to each and similarly placed.

SUPPLEMENTARY EXERCISES

§ 309

Ex. 433. Construct a triangle whose sides are 6, 8, and 10; then construct a similar triangle whose side homologous to 8 is 5.

Ex. 434. Divide a line 10cm long internally in extreme and mean ratio.

Ex. 435. The median from the vertex of a triangle bisects every line drawn parallel to the base and terminated by the sides, or the sides produced.

Ex. 436. Two circles intersect at A and B, and at A tangents are drawn, one to each circle, to meet the circumference of the other in C and D respectively; BC, BD, and AB are drawn. Prove that BD is a third proportional

to BC and AB.

Ex. 437. The diameter AB of a circle whose center is O is divided at any point C, and CD is drawn perpendicular to AB, meeting the circumnference in D; OD is drawn, and CE perpendicular to OD. Prove that DE is a third proportional to AO and DC.

Ex. 438. In the triangle ABC, AD is the median to BC; the angles ADC and ADB are bisected by DE and DF, meeting AC and AB in E and F respectively. Then, FE is parallel to BC.

Ex. 439. A secant from a given point without a circle is 1 ft. 6 in. long, and its external segment is 8 in. long. Find the length of a tangent to the circle from the same point.

Ex. 440. The radius of a circle is 6 in.

What is the length of the tan

gents drawn from a point 12 in. from the center?

Ex. 441. If the tangent to a circle from a given point is 2m and the radius of the circle is 15dm, find the distance from the point to the circumference.

Ex. 442. If from the vertex D of the parallelogram ABCD a straight line is drawn cutting AB at E and CB produced at F, prove that CF is a fourth proportional to AE, AD, and AB.

Ex. 443. If the segments of the hypotenuse of a right triangle made by the perpendicular from the vertex of the right angle are 6 in. and 4 ft., find the length of the perpendicular and the length of each of the sides about the right angle.

Ex. 444. Find the length of the longest and of the shortest chord that can be drawn through a point 7 in. from the center of a circle whose radius is 19 in.

Ex. 445. If the greater segment of a line divided internally in extreme and mean ratio is 36 in., what is the length of the line?

Ex. 446. The shorter segment of a line divided externally in extreme and mean ratio is 240dm. Find the length of the greater segment in meters.

Ex. 447. Find the shorter segment of a line 12dm long when it is divided internally in extreme and mean ratio. When it is divided externally in extreme and mean ratio.

Ex. 448. The tangents to two intersecting circles drawn from any point in their common chord produced are equal.

Ex. 449. If the common chord of two intersecting circles is produced, it will bisect their common tangents.

Ex. 450. ABC is a straight line, ABD and BCE are triangles on the same side of it, having angle ABD equal to angle CBE and AB: BC= BE: BD. If AE and CD intersect in F, triangle AFC is isosceles.

Ex. 451. If in the triangle ABC, CE and BD are drawn perpendicular to the sides AB and AC respectively, these sides are reciprocally proportional to the perpendiculars upon them; that is, AB : AC = BD : CE.

Ex. 452. ABCD is a parallelogram. If through O, any point in the diagonal AC, EF and GH are drawn, terminating in AB and DC, and AD and BC respectively, EH is parallel to GF.

Ex. 453. Lines are drawn from a point P to the vertices of the triangle ABC; through D, any point in PA, a line is drawn parallel to AB, meeting PB at E, and through E a line parallel to BC, meeting PC at F. If FD is drawn, triangle DEF is similar to triangle ABC.

Ex. 454. If two lines are tangent to a circle at the extremities of a diameter, and from the points of contact secants are drawn terminated respectively by the opposite tangent and intersecting the circumference at the same point, the diameter is a mean proportional between the tangents.

Ex. 455. AB and AC are secants of a circle from the common point A, cutting the circumference in D and E respectively. Then, the secants are reciprocally proportional to their external segments; that is, AB : AC = AE: AD.

SUGGESTION. Draw CD and BE, and refer to § 322, 6, b.

Ex. 456. AB and CD are two chords of a circle intersecting at E. Prove that AE: DE CE: BE.

=

Ex. 457. Two secants intersect without a circle. The segments of one are 4 ft. and 20 ft., and the external segment of the second is 16 ft. Find the length of the second secant.

Ex. 458. From a point without a circle two secants are drawn, whose external segments are respectively 7dm and 9dm, the internal segment of the latter being 13dm. What is the length of the first secant?

Ex. 459. The segments of a chord intersected by another chord are 7 in. and 9 in., and one segment of the latter is 3 in. What is the other segment?

Ex. 460. Two secants from the same point without a circle are 24dm and 32dm long. If the external segment of the less is 5dm, what is the external segment of the greater?

Ex. 461. Through a point 7m from the circumference of a circle a secant 28m long is drawn. If the internal segment of this secant is 17m, what is the radius of the circle ?

Ex. 462. If from any point in the diameter of a circle produced a tangent is drawn and a perpendicular from the point of contact is let fall on the diameter, the distances from the point without the circle to the foot of the perpendicular, the center of the circle, and the extremities of the diameter are in proportion.

SUGGESTION. Draw the radius to the point of contact.

Ex. 463. If the sides of a triangle are respectively 1.5Dm, .12Hm, and 10m long, what are the segments into which each side is divided by the bisector of the opposite angle?

Ex. 464. If an angle of one triangle is equal to an angle of another, and the perpendiculars from the vertices of the remaining angles to the sides opposite are proportional, the triangles are similar.

SUGGESTION. Refer to § 322, 6, c and e.

Ex. 465. If two circles are respectively 6 in. and 3 in. in diameter and their centers are 10 in. apart, find the distance from the center of the smaller one to the point of intersection of their common exterior tangent with their line of centers produced.

Ex. 466. Two intersecting chords of a circle are 38 ft. and 34 ft. respectively; the segments of the first are 8 ft. and 30 ft. Find the segments of the second.

Ex. 467. What is the length of a chord joining the points of contact of the tangents drawn from a point 13 in. from the center of a circle whose radius is 5 in. ?

Ex. 468. Chords AB and CD of a circle are produced in the direction of B and D respectively to meet in the point E, and through E the line EF is drawn parallel to AD to meet CB produced in F. Prove that EF is a mean proportional between FB and FC.

Ex. 469. AB is a diameter of a circle, and through A any straight line is drawn to cut the circumference in C and the tangent at B in D. Prove that AC is a third proportional to AD and AB.

Ex. 470. From any point in the base of a triangle straight lines are drawn parallel to the sides. Prove that the intersection of the diagonals of every parallelogram so formed lies in a line parallel to the base of the triangle.

Ex. 471. If E is the middle point of one of the parallel sides DC of the trapezoid ABCD, and AE and BE produced meet BC and AD produced in F and G respectively, prove that GF is parallel to AB.

Ex. 472. If a line tangent to two circles cuts their line of centers, the segments of the latter are to each other as the diameters of the circles.

Ex. 473. The bisector of the vertical angle C of the inscribed triangle ABC cuts the base at D and meets the circumference in E. Prove that

Ex. 474. Through any point A of the circumference of a circle a tangent is drawn, and from A two chords, AB and AC; the chord FG parallel to the tangent cuts AB and AC in D and E respectively. Prove AB: AE=AC: AD. Ex. 475. The greatest distance of a chord 8 ft. in length from its arc is 4 in. Find the diameter of the circle.

Ex. 476. If two circles are tangent externally, their common exterior tangent is a mean proportional between the diameters of the circles.

SUGGESTION. Draw radii to the points of contact, draw the common interior tangent to intersect the common exterior tangent, and connect the point of intersection with the centers.

Ex. 477. The perpendicular from any point of a circumference upon a chord is a mean proportional between the perpendiculars from the same point upon the tangents drawn at the extremities of the chord.

SUGGESTION. Draw lines from the given point to the extremities of the chord, and refer to § 322, 6, c.

Ex. 478. From a point A tangents AB and AC are drawn to a circle whose center is O, and BD is drawn perpendicular to CO produced. Prove that BD is a fourth proportional to AC, CD and CO.

SUGGESTION. Draw AO and BC.

Ex. 479. From a point E in the common base of two triangles ABC and ABD, straight lines are drawn parallel to AC and AD, meeting BC and BD at F and G respectively. Prove that FG is parallel to CD.

Ex. 480. If tangents to a circle are drawn at the extremities of a diameter, the radius is a mean proportional between the segments of any third tangent intercepted between them and divided at its point of tangency.

SUGGESTION. Draw lines to form a right triangle, having the third tangent for its hypotenuse and a vertex at the center.

BOOK V

AREA AND EQUIVALENCE

323. The amount of surface in a plane figure is called its Area. A surface is measured by finding how many times it contains some given square which is taken as a unit of measure.

The ordinary units of measure for surfaces are the square inch, the square foot, the square centimeter, the square decimeter, etc. Suppose that the square M is the unit of measure, and that ABCD is the rectangle to be meas

ured.

D

By applying M to ABCD it is evident that the rectangle may be divided into as many rows of squares, each equal to M, as the A

side of M is contained times in

B

M

the altitude of ABCD; that in each row there are as many squares as the side of M is contained times in the base of ABCD; and therefore, that the product of the numerical measures of the base and altitude of ABCD is equal to the number of times that M is contained in ABCD.

In this case the side of M is contained 4 times in AD and 6 times in AB; consequently, M is contained 24 times in ABCD; that is, the rectangle contains 24 square units.

Therefore, if the side of a square is a common measure of the base and altitude of a rectangle, the product of the numerical measures of the base and altitude expresses the number of times that the rectangle contains the square, and is the numerical measure of the surface, or the area of the rectangle.

324. For the sake of brevity, the product of the base and altitude is used instead of the product of the numerical measures of the base and altitude.