(32.) Having given the sides of a triangle equal to 375.49, 289.63, and 231.19, find its three altitudes.

(33.) If the sides of a triangle are 27.931 m., 2175.4 cm., and 296.53 dcm., what are the lengths in feet of (1) the altitude upon the greatest side, and (2) the segments into which it divides that side?

Hint.—After finding the altitude, the segments can easily be found by logarithms, since (§ 318) x = Va2 — y2 = √(a − y)(a+1').

(34.) Compute the medians of a triangle whose sides are a, b, and c.

Solution. In the triangle CRP, m2 = x2+y3. (1))

c3 __ —

Transposing, 2(x2+y3)=a2+ b2 — C.2 = 2(a2 + b2) — c2

- =

2

2(a2+b2) —c3 ̧

2

x2 + y2 =

4

The other medians are §√2(ba +c2)—a2 and ‡ √2(c2 + a2) — b3. (35) Having given the three sides of a triangle equal to 3, 5, and 7, find its three medians.

(36.) If two sides and one of the diagonals of a parallelogram are respectively 24, 31, and 28, what is the length of the other diagonal?

(37.) In a triangle whose sides are a, b, and c, compute the bisector of the angle opposite c.

Solution. Circumscribe a circle about the triangle, produce the bisector to meet the circumference, and draw BR. Then, in the triangles BCR and CPA, the angle R equals the angle and angle BCR equals the angle PCA. 8 201

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

8320

y = √ ab(1 (a + b)2

c2

(a+b)*

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.

6*

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 altitudes are equal.