In Exs. 13-18 carry the root to two decimal places only.

13. 2.

14. 5.

15. 7.

16. 8.

17. 11.

18. Find the side of a square containing 1 acre (160 sq. rd.).

254. Hypotenuse. In a right-angled triangle the side. opposite the right angle is called the hypotenuse.

255. Square on the hypotenuse. If a floor is made up of triangular tiles like this, it is easy to mark out a right-angled triangle. In the figure it is seen that the square on the hypotenuse contains 8 small triangles, while each square on a side contains 4 such triangles. Hence

The square on the hypotenuse equals the sum of the squares on the other two sides.

4

3

256. This is true for any right-angled triangle. We see that if the 4 triangles, 1 + 2 + 3 + 4, are taken away from this figure, there remains the square on the hypotenuse. But if we take away the 2 shaded rectangles, which equal the 4 triangles, there remain the squares on the two sides. Therefore the square on

the hypotenuse must equal the sum of these two squares.

257. Illustrative problem. If AB = 12, and AC = 9, how

1. How long is the diagonal of a hall 51 ft. by 68 ft.? 2. How long is the diagonal of a square containing 4 sq. ft.? (Two decimals.)

3. The two sides of a right-angled triangle are 20 in. and 30 in. Find the hypotenuse. (Two decimals.)

4. The two sides of a right-angled triangle are 57 in. and 76 in. Find the hypotenuse.

5. What is the direct distance from the cornice of a 100-ft. building to a spot 75 ft. from the foot?

6. Find the length of the hypotenuse when the sides are 321 in. and 428 in.; 40 in. and 75 in.; 72 ft. and 135 ft.

7. A telegraph pole is set perpendicular to the ground, and a wire is fastened to it 18 ft. from the ground, and then to a stake 13 ft. 6 in. from the foot of the pole, so as to hold it in place. How long is the wire?

Arm 27'6"

22'

8. A derrick for hoisting coal has its arm 27 ft. 6 in. long. It swings over an opening 22 ft. from the base of the arm. How far is the top above the opening?

Reversing the procedure in § 255, the square on either side equals the difference of what squares?

Find to two decimal places the hypotenuse of each of the right-angled triangles of which the sides are here given:

9. 35 ft., 26 ft.

11. 10 rd., 13 rd.

13. 4.5 in., 7.2 in.

10. 81 ft., 35 ft.

12. 421 ft., 631 ft.

14. 6.25 in., 7.5 in.

It is a good exercise to make up problems like those in Exs. 15–19. 15. A room is 16 ft. long, 12 ft. wide, and 9 ft. high. How far is it from an upper corner diagonally through the room to the opposite lower corner? (Two decimals.)

First draw the picture. Then find the hypotenuse on one wall. Then find the hypotenuse required.

16. How long is the diagonal of a cube whose volume is 8 cu. in.? (Two decimals.)

First find the edge. Then proceed as in Ex. 15.

17. A school flag pole is broken by the wind 16 ft. from the ground. The two pieces hold together, and the top of the pole touches the ground 30 ft. from the base. Find the length of the pole.

18. If I start to row directly across a stream, in the direction AC, at the rate of 4 mi. an hour, and if the stream

B

3

carries me in the direction CB at the rate of 3 mi. an hour, my course will really be AB, the result of these two motions. Suppose I row at the rate of 4.5 mi. per hour, and the stream

flows 6 mi., what is my rate of progress?

19. Suppose I walk across the deck of a steamer at the rate of 4 mi. per hour, while the boat moves at the rate of 8 mi. per hour, at what rate am I moving? Answer to two decimal places. (Draw a plan to scale.)

258. Helpful approximations. Because of their frequent use, it is helpful to learn the following approximations : √5 = 2.236 √10 = 3.162

√2=1.414

√3 = 1.732

259. Illustrative problem. square whose side is 7 in.?

What is the diagonal of a

1. The square on the diagonal (hypotenuse) is 49 sq. in. + 49 sq. in. = 2 × 49 sq. in.

1. From § 259 write out a rule for finding the diagonal of a square by multiplying its side by a certain number. Find the diagonals of squares with sides here given : 2. 19.2 in. 3. 32.8 in.

4. 683 ft.

5. 750 rd. 6. If the diamond of a baseball field is a square 90 ft. on a side, how far is it from the first base directly across to the third?

7. A gate 3 ft. high and 6 ft. wide is to be braced by a stick fastened diagonally across it. How long is the stick? Notice that √45 = √9 × 5

=

3 × √5.

8. How far is it from one lower corner of a room directly to the opposite upper corner, the dimensions of the floor being 12' x 16' and the room being 8 ft. high?

Find the diagonals of squares with areas here given: 10. 256 sq. ft. 11. 6.25 sq. ft.

9. 81 sq. ft.

Find the sides of squares with diagonals here given :

12. 14.14 in.

13. 4.242 in.

14. 7.07 ft.

260. How to find a cube root. Find

2197.

This subject may be omitted without interfering with the work that

Imagine a cube containing 2197 cubic units, as in A, where of course we cannot see all of the small cubes.

The greatest cube of tens in 2197, or A, is 1000 (B), for 203 = 8000, and this is greater than 2197.

Taking away 103, or 1000, as marked off in C, we have left 1197, as shown in D.

We may now lay D lengthwise, as in E.

E