B

A right triangle is a triangle having one right angle. The two other angles of a right triangle are acute angles. The two sides which form the right angle are called the "legs" of the triangle, and the remaining side is called the "hypotenuse"; thus, in the right triangle ABC shown in Figure 1, the sides AB and AC are the legs and BC is the hypotenuse.

A

(Figure 1) Over two thousand years ago, a Greek mathematician named Pythagoras discovered the fact that when squares are drawn on all three sides of a right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the two other sides. This rule is known as the Pythagorean theorem.

To prove that this is so, we need only to make three cardboard squares to correspond with the three sides of a right triangle as shown in Figure 2. Taking the square which is neither the largest nor the smallest, that is Square A, find the point where its diagonals cross, that being the exact center of the square. Now divide and cut this square into four equal parts by drawing two lines through the center of the square, the first line running parallel to the hypotenuse of the given triangle,

and the second line crossing the first line at right angles, as shown in Figure 3. Now lay Square B and the four parts of Square A on Square C as indicated in Figure 3.

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It naturally follows, that since the square on the hypotenuse (C = 25 sq. in.) is equal to the sum of the squares on the two other sides (A 16 sq. in. + B = 9 sq. in.) then C - B = A, and CA B; in other words, the square on the hypotenuse minus the square on either leg equals the square on the other leg.

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Since the length of one side of a square is equal to the square root of the area, we can find the length of any side of a triangle when we know the area of the square on that side by extracting the square root of such area. Thus, in Figure 2, if we know that the area of A is 16 sq. in., we know that the length of that side is 16 in., or 4 in.; in like manner, the area of B is 9 sq. in., therefore, the length of that side is √9 in., or 3 in.; and the area of C is 25 sq. in., therefore, the hypotenuse is √25 in., or 5 in.

Count the blocks in the three squares shown in Figure 2 above? Is it true?

Make one of your own and use 6" and 8" as the legs to find the hypotenuse. Prove by blocking it as he did. Now prove by cutting it and fitting the parts.

The sign is often used to indicate a triangle; ▲ for triangles.

Exercise 32-Oral.

The children of one row are to ask the following questions of the others:

1. Referring to the triangle shown in Figure 2, state the length of each of the two legs. Of the hypotenuse.

2. What kind of a triangle is this? Why?

3. What is the ratio of a square constructed on the hypotenuse of a right triangle to the sum of squares constructed on the two legs?

4. State the rule covering the theorem discovered by Pythagoras regarding squares constructed on the sides of right triangles.

5. Again referring to Figure 2, how can we find the area of Square C when we know the areas of A and B? How can we find A when we know How can we find B when we know

B and C?

A and C?

6. How can we find the square on either leg of a right triangle when the squares on the hypotenuse and on the other leg are known?

7. If we know the length of any side of a A, how do we find the square on that side?

8. If we know the square on any side of a ▲, how do we find the length of that side?

9. If the square on the hypotenuse is 64 sq. in., what is the length of the hypotenuse?

10. If the squares on the two legs of a right triangle are respectively 36 sq. in. and 64 sq. in., what is the area of the square on the hypotenuse? What is the length of the hypotenuse? What is the length of each of the two legs?

Exercise 33-Written.

1. The legs of a right triangle are respectively 15 ft. and 20 ft.; what is the length of the hypotenuse? 2. The hypotenuse of a right A is 95 ft. and one of

the legs is 57 ft.; what is the length of the other leg?

3. The hypotenuse of a right triangle is 221⁄2 in. and one of the legs is 13 in.; what is the length of the other leg?

4. The legs of a right ▲ are respectively 6 ft. and 8 ft. long; what is the length of the hypotenuse?

(Make a chalk mark on the floor 8 feet from a wall, and make one on the wall 6 feet from the floor; prove, by using a string of the length corresponding to the hypotenuse of this triangle, that the wall is perpendicular.)

5. Using the principle outlined in Question 4, open a door so that it forms an exact right angle with the wall.

6. The legs of a compass are 5 in. long; what is the distance from the point of one leg to the point of the other leg when the compass is opened to a right angle?

7. A smoke-stack 40 ft. high is to be held rigid by 4 wires running from the top of the stack to points 30 ft. from the bottom of the stack; allowing 10 ft. for fastening each of these 4 wires, what is the total length of the wire required to anchor this stack?

8. A baseball diamond is 90 ft. square; how far is it from first base to third base?

9. What is the length of the diagonal of a 20-ft.

square?

10. The area of a square lot is 15,625 sq. ft.; what is the length of its diagonal?

11. Find the longest line in a rectangle 20 yd. long and 12 yd. wide.

12. Find the diagonal of the ceiling of a room 40' long and 35' wide.

13. Find the length of the line from the upper corner of a room to the lower corner diagonally opposite; the room measurements are: length, 45 ft.; height, 15 ft.; width, 32 ft.

(Suggestion: Get the diagonal of the ceiling first.)

14. Find the volume of a cube whose entire surface is 486 sq. in.

LESSON 17

Isosceles and Equilateral Triangles

A triangle having three equal sides is an "equilateral" triangle.

A triangle having two equal sides is an "isosceles" triangle; therefore, every equilateral triangle is also isosceles.