Substituting values for x and (c-x) from (2) and (3)

yz=(a+b)2

$320

abc

Subtracting (4) from (1) y2= ab —

c2

(a+by=ab

y =√ ab(1. (a + b)2 2

This result may be factored and arranged for logarithmic computation as follows:

(38.) If the sides of a triangle are 219.57, 178.35, and 153.94 ft., find the length of the bisector of the angle opposite the greatest side.

(39.) If the sides of a triangle are a, b, and c, find the radius of the circumscribed circle.

Solution. Suppose the diameter CS of the circle to be drawn from C. Draw SA and the altitude CP.

Then in the right triangles CSA and CBP the angle CAS is equal to the angle P (§ 202), and the angle S is equal to the angle B.

Therefore the triangles are similar, and

8 201

(40.) If the sides of a triangle are 125.76, 119.53, and 98.991

ft. in length, find the radius of the circumscribing circle expressed in meters.

PLANE GEOMETRY

BOOK IV

AREAS OF POLYGONS

374. Def.-The area of a surface is the ratio of that surface to another surface taken as the unit.

The unit surface may have any size or shape, but the most common and convenient unit is a square having its side equal to the unit of length, as a square inch, a square mile, etc.

375. Def.-Equivalent figures are figures having equal

areas.

We may observe (1) figures of the same shape are similar.

(2) figures of the same size are equivalent.

(3) figures of the same shape and size are equal.

376. Defs.-The bases of a parallelogram are the side upon which it is supposed to stand and the opposite side.

The altitude is the perpendicular distance between the bases.

PROPOSITION I. THEOREM

377. Two rectangles having equal bases and equal alti tudes are equal.

B

B

D'

GIVEN-two rectangles, AC and A'C', having equal bases, AD and A'D', and equal altitudes, AB and A'B',

Hence the rectangles coincide throughout and are equal. § 15

PROPOSITION II.

THEOREM

Q. E. D.

378. Two rectangles having equal bases are to each other as their altitudes.

B

A

GIVEN two rectangles AC and A'C', having equal bases, AD and A'D'.

B

A

CASE I. When the altitudes, AB and A'B', are commensurable.

Suppose AO, the common measure of the altitudes, is contained in AB three times and in A'B' twice.

Through the several points of division draw parallels to the bases.

The rectangle AC will be divided into three rectangles and A'C' into two, all five of which will be equal.

$377

$180

Ax. I

rect. A'C'

CASE II. When the altitudes, AB and A'B', are incommen