To find the square root of a mixed decimal, such as 333.4276, begin at the decimal point and separate into periods of two figures each. Thus, 3'33.42'76.

EXAMPLE 1. Find the square root of .00498436.

In this solution it was necessary to find the trial divisor for the fourth figure in order to be sure that this fourth figure is less than 5.

Exercise 32

1. Find the first figure which is not zero in the square root of each of the following: .04; 40; 4; .0009; .009; .1; .03; .003; 3; .16; 1.6; .016; .0016.

Find to the nearest .0001 the square root of each of the following:

Reduce these common fractions to decimals, then find their square roots to the nearest .0001.

19. How long is one side of a square whose area is one

acre?

20. Find the side of a square whose area is .3 of an acre. 21. A certain rectangle is three times as long as it is wide and its area is 15 sq. ft. What is its width to the nearest .01 ft.?

HINT. Divide the rectangle into squares and find the side of one of the squares.

32. The square root of a common fraction.

Multiply by itself.

Square ; ; . Make a rule for squaring a fraction. Find the square root of . Check the result by squaring it. Find the square root of 4. Check the result.

Make a rule for finding the square root of a common fraction.

The rule is easily applied if both the numerator and the denominator are square numbers. In other cases it is simpler to reduce the common fraction to a decimal and then extract the square root.

Rule. The square root of a common fraction is found by taking the square root of both numerator and denominator.

Exercise 33

Find the square root of each of the following:

7. 301. 8. 694.

9. 10.

10. 117.

6. 27. 11. Find in two ways the square root of to four decimal places. First reduce to a decimal, then extract the root; next extract the square root of both terms and reduce the result to a decimal. Which is the simpler way?

12. Find in two ways the square root of to four decimal places.

HINT. First multiply both terms of the fraction by 7.

13. Find in two ways the square root of to four decimal places.

1. Define square root.

Exercise 34

2. State the principle for finding the square of the sum of two numbers.

3. Use this principle in finding the following squares : (t+w)2; (60+8)2; (7+1)2; (40+a)2.

4. Find the square root of each of the following squares: t2+2tw+w2; 602+2 60·8+82; 72+2·7·1+(1)2; 402+2·40. a +a2.

Find the following correct to .01:

Name the first figure which is not zero in the square root

29. Find correct to .0001, the side of a square whose area is 2 sq. ft.

30. The area of one face of a cube is .0006 m2. Find the length of one edge of the cube correct to .001m.

31. It is shown in geometry that the area, A, of a triangle whose sides are a, b, and c, is given by the formula

A=√S(S− a) (S—b)(S—c),

where S=a+b+c. Find the area of a triangle whose sides

2

are 14, 10, and 8.

32. Find the number of acres in a triangular field whose sides are 18 rd., 28 rd., and 32 rd.

33. Relation between the sides and the hypotenuse of a right triangle. The side of a right triangle opposite the right angle is called the hypotenuse.

The C

other two sides are usually referred to as the sides.

In Figure 10, AB is the base and BC is the altitude of the right triangle.

Construct a right triangle with sides 3 in. BL and 4 in. Measure the hypotenuse. If your

FIG. 10

A

measurement and construction are accurate the hypotenuse will be found to be 5 in.

Construct a right triangle with sides 6 in. and 8 in. Measure the hypotenuse. How long is it?

Construct another right triangle with sides 5 in. and

FIG. 11

12 in. How long is the hypotenuse?

Construct squares on the three sides of a right triangle whose sides are 3 in., 4 in., and 5 in., as in Figure 11. Since 9+16=25,

the square on the hypotenuse of this right

triangle equals the sum of the squares on the other two sides.

Test to see if this is true for the other two triangles that you have constructed. These examples illustrate the following important fact known as the

Pythagorean theorem. The square on the hypotenuse of a right triangle equals the sum of the squares on the other two sides. It is proved in geometry that this theorem is true for all right triangles. It is one of the most important theorems

in geometry and is used in solving many practical problems.

If h is the hypotenuse, a the altitude, and b the base of a right triangle as in Figure 10, then the Pythagorean theorem may be stated by the

By taking the square root of each side of this equation we get

h=√a2+b2.

This gives a formula for finding the hypotenuse of a right triangle when the two sides are known. Let the pupil state this formula as a rule.

Exercise 35

Find the hypotenuse of a right triangle, given the sides as follows:

1. 15 in., 20 in.

2. 12 in., 35 in.

3. 10 in., 10 in.

4. 100 ft., 65 ft.

5. From the formula h2=a2+b2 we wish to get a formula to find the altitude of a right triangle when the base and the hypotenuse are known. From h2=a2+b2 find a2. Now find a. State as a rule the formula you have found.

6. Obtain a formula for finding the base of a right triangle when the hypotenuse and the altitude are given. State this formula as a rule.

7. Find a if h=50 ft. and b=14 ft.

8. Find b if a=21 and h=35.

9. Find the base of a right triangle whose altitude is 65 ft., and hypotenuse 98 ft.

10. Find the altitude of a right triangle whose hypotenuse is 56.56 rods, and base 40 rods.