272. Illustrative Example.

Find the area of a triangle with

a base of 16 rd. and an altitude of 20 rd.

EXPLANATION. The number of square rods in the area of a rectangle with a base of 16 rd. and an altitude of 20 rd. is 20 × 16 = 320. The number of square rods in the area of a triangle of the same base and alti16 × 20 tude is as many as in the rectangle, that is, 2

160. Ans. 160 sq. rd.

1. There are how many square feet in a triangular platform having a base of 91 ft. and an altitude of 10 ft.?

2. Find the area of a triangle that has a base of 20 ft. and an altitude of 13 ft.

3. Find the area of a triangle whose base is 121 ft. long and altitude 4 ft.

4. Find the area of a triangle whose base is 10 yd. 2 ft. long and altitude 3 yd. 1 ft.

5. What must be the height of a triangle that contains 2250 sq. ft. and whose base is 180 ft. in length?

6. How many square feet are there in a triangular flower bed whose base is 19 ft. and the height 9 ft. 6 in. ?

7. Mary is to crochet a half square breakfast shawl, each of its two equal sides being 14 yd. long. How many square feet will she crochet?

8. Find the surface of an octagonal spire, each of whose eight faces is 191 ft. at the base and 513 ft. high.

9. Formulate a rule for finding the area of a triangle when the base and the altitude are given.

10. Formulate a rule for finding either the base or altitude of a triangle, the area and either the base or altitude being given.

273. To find the area of a rhombus and of a rhomboid. What is the area of each of these rectangles?

It will be seen by the diagram that the rhombus ABCD is equal to the rectangle EBCF of the same base and altitude.

If the shaded part, NPO, of the rhomboid LMNO is cut off, it will be found to fill exactly MRL, thus giving what kind of a figure?

Compare the base of the rhomboid with the base of the rectangle. Compare the altitudes.

This proves that the area of a rhomboid is equal to that of a rectangle of the same base and altitude. The same principle may be applied to finding the area of a rhombus.

1. How many square feet are there in a rhomboid whose base measures 41 ft. and whose height is 13 ft. ?

2. The base of a rhombus is 14 ft. and its altitude 11 ft. What is its area?

3. What is the area of a parallelogram whose base is 36 ft. and altitude 16 ft. ?

3 in.

plan.

95 rd.

85 rd.

274. To find the area of a trapezoid.

If you draw a perpendicular from C to the base AB, you will have a rectangle and a triangle. Find the area of each and add the results, thus obtaining the area of the trapezoid ABCD.

1. What is the area of a trapezoid, the longer of the two parallel sides being 120 ft., the shorter 110 ft., and the altitude 75 ft.?

2. A plank 25 ft. long is 11 ft. wide at one end and 1 ft. wide at the other. What is the area of one side of this plank?

3. Find the number of acres in the farm of which this is a

100 rd.

80 rd.

Find the area in acres of the field (see p. 181) used for:

275. ABC is a right-angled triangle whose sides are 3 in., 4 in., and 5 in. respectively. The square formed upon the hypotenuse, AC, contains how many square inches? The square formed upon the base, BC, contains how many square inches? The square formed upon the perpendicular, AB, contains how many square inches? How does the number of square inches upon AB and BC together compare with the number of square inches upon the hypotenuse, AC?

The length of the hypotenuse of a right-angled triangle equals the square root of the sum of the squares of the other two sides. 276. Illustrative Example.

8 in.

6 in.

The base of a right-angled triangle is 8 in. and the perpendicular 6 in. What is the hypotenuse?

82=64

62 = 36

V10010 Therefore the hypotenuse is 10 in. Ans.

In the diagram (page 182) compare the number of squares upon the base BC with the number upon AB and AC. It will be found that the number upon BC is the same as the difference in number between those on AC and AB. This will also hold true in comparing the number of squares upon the perpendicular AB with the difference in number of those upon AC and BC. Therefore,

The square root of the difference of the squares of the hypotenuse and the other side will give the base or the perpendicular.

1. What is the perpendicular if the base of a triangle is 8 in. and the hypotenuse 10 in.?

2. Find the hypotenuse if the base is 21 in. and the perpendicular 20 in.

3. What must be the length of a ladder to reach the top of a house 24 ft. high, the foot of the ladder being placed 18 ft. from the house?

4. If a ladder 26 ft. long is placed with its foot 10 ft. from a wall, how high up the wall will it reach?

5. Two boys start from the corner of a square field and walk along the two sides. How far apart will they be when one has walked 130 yd. and the other 144 yd. ?

6. Two ships sail from a harbor, one 56 mi. due east, and the other 90 mi. due south, in a day. How far apart are they at the end of the day?