11. The sides of a right triangle are 15 in. and 39 in. What is the sum of the squares on these sides? What is the square on the hypotenuse? How long is the hypotenuse?

12. Find the hypotenuse of a right triangle whose sides are 12 and 26.

13. Find the diagonal of a rectangle whose dimensions are 20 in. and 30 in.

14. A baseball diamond is a square 90 ft. on a side. The catcher stands 3 ft. behind home plate and throws to second base. How far does he throw?

15. What is the longest straight line that can be drawn on a piece of paper 8 in. by 11 in.?

16. The hypotenuse of a right triangle is 60 ft. and one side is 36 ft. What is the square on the hypotenuse? What is the square on the given side? What is the square on the other side? Find the length of the other side.

17. A rope 100 ft. long is attached to the top of a flagpole and reaches to a point on the ground 80 ft. from the foot of the pole. How high is the pole?

18. A ladder 20 ft. long is set with its foot 8 ft. from the base of a vertical wall and with its top resting against the wall. How high up the wall

20. A house is 18 ft. wide and the ridgepole is 9 ft. above

the plate. Find the length of the rafter if it reaches 1 ft. beyond the plate.

18

Ridgepole

21. Two fence posts each 4 ft. high are 8 ft. apart. How long must a brace be made to reach from the bottom of one post to a point 6 in. below the top of the other?

22. A kite string is 400 ft. long. The boy who holds one end of the string is 200 ft. from a point directly under

the kite. How high is the kite? Make no allowance for slack in the string.

23. The pitcher in a baseball game catches a batted ball while standing at the point of intersection of the diagonals of the diamond. The batter and the pitcher both run to first base. How much farther does the batter have to run than the pitcher?

24. A room is 30 ft. long, 20 ft. wide, and 12 ft. high. Find the length of a wire that reaches from one upper corner to the opposite lower corner.

25. How many times the side is the diagonal of a 1-inch square? Answer the same question for a 3-inch square; for a 20-inch square; for a square whose side is a.

26. By what must the side of a square be multiplied to get the diagonal? If given the diagonal, how can you find the side? Make formulas which give the answers to these questions, using d for the diagonal and s for the side.

27. The side of an equilateral triangle ABC, Figure 14, is 10 in. Find the altitude, knowing that D, the foot of the altitude, bisects the base AB. Find the answer correct to

.001 sq. in. Show that the altitude of this equilateral triangle is .866s, where s is the side. Find the altitude of an equilateral triangle whose side is 100 ft.;

30 ft.; 80 rods.

28. The last exercise gives the formula h.866s, where h is the altitude of an equilateral triangle and s is the side. Find the altitude of an equilateral triangle whose side is 450 ft.; 65 ft.; 320 rods.

if

29. How can you find the side of an equilateral triangle you know the altitude? State your answer as a formula. Find the side of an equilateral triangle whose diagonal is 173.2 ft.; 13.856 ft.; 26 rods.

30. Find the area of an equilateral triangle whose side is 10 in.

31. Explain how this formula for the area of an equilateral triangle is made: A=hs = .866 8.433s. Here A repre

2

sents the area, h the altitude, and s the side.

32. Find the area of an equilateral triangle whose side is 16 in.

33. Show that a regular hexagon is made up of six equilateral triangles. Find the area of a regular hexagon whose side is 4.5 in.

B

FIG. 15

34. Finding square roots graphically. In the right triangle ABC (Figure 15), AB=3, BC=4, and therefore AC=5.

The length of AC may be tested A by using the compasses. Open the compasses the distance AC

and see if this distance is 5.

In the right triangle DEF, DE=1, EF=1,

1

E

FIG. 16

and therefore DF=√2. If DE=a and EF-a, what is the

length of DF?

1. Construct the square root of 7. See Figure 19. 2. Construct the square root of 13. Notice that 13=9+4.

3. Construct the square root of 20; of 10; of 41; of 61. Check the answers by measuring with the compasses.

M

3

N

FIG. 19.

4. If the hypotenuse of a right triangle is 6 and the base 3, what is the altitude? Make such a triangle and thus construct √27.

5. Construct √45.

HINT. Make the hypotenuse 7. What then shall the base be made?

6. I construct a square with side 10. I then construct a second square with a side equal to the diagonal of the first. What is the area of the first square? Of the second square? The area of the second square is how many times the area of the first?

7. I construct a square with side a. I then construct a second square with a side equal to the diagonal of the first. What is the area of the first square? Of the second square? The area of the second square is how many times the area of the first?

8. I wish to construct a square with an area twice as large as the area of a given square. How should I proceed?

9. A gardener wishes to lay out a square flower bed twice as large as another square one whose side is 16 ft. Can you tell him how long the side of the new bed should be made?

10. The distance that a sailor can see from the top of a mast is given approximately by the formula d=1.22√h, where d is the distance in miles and h is the height in feet of the mast above the water. How far can a sailor see if he is 30 ft. above the water? If he is 80 ft. above the water?

11. The velocity v of the water at the bottom of a stream is calculated by the formula v = V+1−2√V, where V is the velocity of the surface. Find v if V is 2 miles an hour; if V is 100 ft. a minute.

12. T=2√ Find T if 1=36 and g=32.2.

g

13. If a body falls h feet, its velocity, V, is given by the formula V√2gh, where g=32.2. In this formula the resistance of the atmosphere is neglected. Find the velocity when it reaches the ground of a ball dropped from a tower 90 ft. high.

14. A pile driver falls 18 ft. How fast is it going when it strikes?

15. A hickory nut falls 40 ft. How fast is it going when it strikes the ground?