44. THE PYTHAGOREAN THEOREM

1. Draw a right triangle with the sides which form the right angle 3 inches and 4 inches respectively.

2. Measure the length of the other

side, or hypotenuse.

3. Draw a square on each of the three sides as base.

4. Compare the square on the hypotenuse with the sum of the squares on the other sides.

Pythagoras proved about 500 B.C. that the fact that we find true here is true for any right triangle, viz. that

The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

5. Carpenters make use of this fact in laying out the foundation for a building, when they want to form a right angle. A line 8 feet long is taken in one direction along which the foundation is to be made. Another line 6 feet long is fastened to one extremity of the first line and moved until a 10-foot rod will just reach the outer extremities of the two lines. Draw such a figure, and show that this gives a right triangle.

6. Use the test in Exercise 5, and find whether the walls of your schoolroom are perpendicular to the floor.

7. If the square on the hypotenuse is 100 sq. in. and on one of the sides 36 sq. in., what is the length of each side of the triangle?

8. Find the diagonal of the ceiling of your room by measuring the length and breadth of the room.

Denoting the hypotenuse by H, the base by B, and the perpendicular by P, when these are abstract numbers representing the number of units in the dimensions, we may state from the above principle the following formula:

H=√B2 + P2

B = √H2— P2

P=√H2- B2

9. Explain the formulæ.

10. If H 15 and P = 14, B = what?

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11. If B 15 and P = 16, H= what?

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12. If H= 25 and B = 20, P

= what? The truth of the Pythagorean theorem may be seen by drawing, or cutting from cardboard, figures like the fol

Let ABC be the right triangle. The square on the hy potenuse AC is equal to the four triangles, 1, 2, 3, and 4, and the small square, 5. Now put 1 and 2 in the position of the figure at the right, and the figure is equal to a square on AB and one on CB.

Problems

1. The base of a right triangle is 48 feet and the perpendicular is 36 feet. What is the hypotenuse?

2. The hypotenuse is 85 feet and the perpendicular is 51 feet. What is the base?

3. The base is 76 feet and the hypotenuse is 95 feet. What is the perpendicular?

4. What is the diagonal of a rectangle 92 ft. long and 69 ft. wide?

5. What is the diagonal of a 30-ft. square?

6. What is the longest line that can be drawn on a sheet of paper 16 inches wide and 20 inches long?

7. What is the diameter of the largest wheel that can be got through a doorway measuring 7 feet by 5 feet?

8. What is the distance between the opposite corners of a field 200 rods long and half as wide?

9. If a window is 18 ft. from the ground, how long must a ladder be to reach to the window if the foot of the ladder is placed 6 ft. out from the building?

10. In decorating a room two ribbons are stretched, connecting the opposite corners. If the room is 30 ft. wide and 40 ft. long, how many yards of ribbon does it take?

11. A baseball diamond is 90 ft. square. from first to third base?

How far is it

12. A derrick is 48 ft. high, and is supported by three steel cables, each reaching from the top of the derrick to a stake in the ground 45 ft. from the foot of the derrick. How much steel cable does it take, allowing 10 ft. for fas tening all three cables?

13. The gable of a house is 24 ft. wide and 12 ft. high from the plate to the ridgepole. How long must the carpenters cut the rafters, if they are to project one foot over the eaves?

14. There are 16 steps to a stairway. The rise of each is 8 in. and the tread 10 in. How long must the timber be cut that runs from one floor to another to support the steps?

45. ISOSCELES AND EQUILATERAL TRIANGLES

A triangle having two equal sides is isosceles. One having all of the sides equal is an equilateral triangle.

ISOSCELES TRIANGLE

EQUILATERAL TRIANGLE

Prove by cutting or measuring that:

(1) The altitude of an isosceles triangle divides the base into two equal parts.

(2) The perpendicular from any vertex of an equilateral triangle to the opposite side divides that side into two equal parts.

Since an equilateral triangle is also isosceles whatever side is taken as base, (2) could have been inferred from (1).

1. If the base of an isosceles triangle is 12 in. and the equal sides 10 in., what is the altitude?

2. Find the altitude of a triangle 10 in. on each side.

3. Find the area of an isosceles triangle whose base is 10 in. and whose equal sides are each 12 in.

4. Find the area of an equilateral triangle each of whose sides is 14 in.

5. A regular hexagon is made up of six equilateral triangles. Study the figure and discover how to inscribe one in a circle.

6. Find the area of a regular hexagon each of whose sides is 10 in.

A REGULAR HEXAGON

46. ADDITIONAL APPLICATIONS OF PYTHAGOREAN

THEOREM

1. Find the volume of a pyramid whose slant height is 10 in. and whose base is 12 inches square.

2. The slant height of a square pyramid is 15 inches, and the side of the base is 10 inches. Find the volume.

3. A square pyramid has a base whose side is 4 ft., and the altitude is 6 ft. Find the slant height.

4. What two lines of an isosceles triangle must be known to find the area? Find the area of an isosceles triangle whose equal sides are each 10 ft. and whose base is 8 ft.

5. The altitude of a pyramid whose base is an equilateral triangle is 8 inches. The sides of the base are each 6 inches. Find the volume.

6. If the slant height of a cone is 12 inches and the diameter of the base 10 inches, find the altitude.

7. Find the altitude of a cone when the diameter of the base is 16 in. and the slant height is 15 in.

8. Find the volume of a cone when the radius of the base is 6 in. and the slant height 10 inches.

9. A pile of grain in the shape of a cone is 12 ft. in diameter at the bottom and the slant height is 9 ft. Find how many bushels it contains. (1 cu. ft. 0.8 bu.)

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