Testing the Table of Square Roots

By use of the table find the square root of:

Find to the nearest hundredth the square root of:

The Principle of Pythagoras. - Pythagoras was one of the greatest of ancient mathematicians. Although a Greek, he lived and taught in Italy, where he died about 500 B.C. He discovered this famous and useful principle: The square on the hypotenuse of a right triangle is equal to the sum of the squares on its legs.

The accompanying diagram illustrates the truth of this principle.

PYTHAGORAS

spectively, the hypotenuse will be 5 inches.

How many square inches in each of the squares shown in the diagram?

Then does 9+16=25?

If the legs are called a and b and the hypotenuse is called c, then a2 + b2 = c2.

Or

=

c2 - a2.

Also a2

= c2 b2 and b2

Indirect Measurement by the Pythagorean Princi

ple.

6

- Frank's cousin John used to visit him often. To do so he went north 8 miles, then east 6 miles. One day the boys were wishing they might go in a straight line and were wondering how much they would save by doing so. One of them drew a diagram like this. Then they saw that the distance they wished to find was the hypotenuse of a right triangle whose legs are 6 miles and 8 miles.

So if it were possible for them to go in a straight line, they would save 4 miles.

Everett had a long reed fishpole which would just lie diagonally on the floor of the porch from one corner to the opposite corner. The

porch without measuring it. This is the way he did it.

Finding the Sides of a Right Triangle by Square Root Using the formulas on page 120 find the value of c when

26. Find the hypotenuse of a right triangle whose legs are 36 and 15 inches.

27. A ladder is set so that it reaches a window 24 feet high. Its foot is 10 feet from the house. How long is the ladder?

28. If a ladder 41 feet long is set with its foot 9 feet from a house, how high will it reach?

29. Find the side of a square lot which contains 4356 square feet.

30. A rectangle is 32 feet long and the length of its diagonal is 40 feet. How wide is it?

31. A boy walked 8 rods north and then 12 rods west. How far in a straight line was he from his starting point?

32. A baseball diamond is a square 90 feet on a side. How far is it in a straight line from the home plate to second base?

33. Two trains leave the same station at the same time on straight tracks at right angles to each other, one going 36 miles an hour and the other 48 miles an hour. How far apart in a straight line will they be at the end of 11⁄2 hours? 34. The side of a certain square is 16 feet. What is the side of a square whose area is twice as large?

35. How much more fence will it require to inclose a rectangular lot 90 feet by 40 feet than it will a square lot having the same area?

36. The altitude of an isosceles triangle is 20 inches; the base is 24 inches. What is the length of one of the equal sides?

Hint: The altitude divides an isosceles triangle into two equal right triangles.

37. Is a triangle whose sides are 3 inches, 4 inches, and 6 inches a right triangle? Why?

38. Find the side of a square field which contains 9741 square feet.

39. Mr. Brown had a rectangular field 72 by 26 rods, and a square field of the same area. Find the number of yards of fence necessary to inclose each field. Which requires the longer fence? How much longer?

40. The area of a circle is 4573 square inches. the side of a square which has the same area?

the perimeter of that square?

What is What is

41. See if you can find the cube root of the following by a plan similar to that used at the bottom of page 117.

(a) 50. Hint: Cube root of 64 = 4; of 27 = 3.

(b) 75.

(c) 100.

(d) 150.

(e) 200.

Speed Test in Finding Square Root

8 right in 10 minutes

Find the square root to two decimal places:

Using the Principle of Pythagoras to Test Triangles. — The principle of Pythagoras enables us to tell whether a triangle is right, obtuse, or acute if we know the length of the sides.

The longest side is called c and the others a and b. Then if c2 = a2 + b2, the triangle is a right triangle.

If c2 is greater than a2 + b2, the triangle is an obtuse triangle.

If c2 is less than a2+ b2, the triangle is acute.