FINDING DISTANCES BY SCALE DRAWINGS 257 Constructing and Finding Angles 1. Draw five acute angles and mark your estimate on each angle. Have one of your classmates measure these angles with a protractor. 2. Draw five obtuse angles and do as in problem 1. 3. Repeat this work with many angles until you are able to estimate angles with a fair degree of accuracy. 4. Draw several triangles. of the angles of each triangle. Measure and find the sum What does this show you? The sum of the angles of a triangle is equal to 180°. 5. Knowing two angles of a triangle how can you find the third? 6. What is the sum of the acute angles of a right triangle? 7. One angle of a triangle is 70°. What is the sum of the other two? 8. Two angles of a triangle are 60° and 40°. Calling the third angle x, form an equation and find x. 9. Find the third angle of a triangle when two of the angles are 65° and 55°. 10. The three angles of an equilateral triangle are equal. How large is each? 11. The three angles of a triangle are represented by x, x + 5, and x + 25. Find the size of each angle. 12. One angle of a triangle is x, the second is twice as large, and the third is three times as large. Find the angles. What kind of triangle is it? Finding Distances by Scale Drawings If a drawing is made accurately to a certain scale, any distances on the drawing may be found by using the same scale for measurement. PROBLEMS A farmer wished to know the distance diagonally across a field 40 rd. wide and 60 rd. long. 60 rd. rd. = He drew a plan of the field, letting 1" = 10 rd. He found that the line AB was 7" long. As 1" 10 rd., 7′′ = 72.5 The distance across the field is 72 rd. To find the distance across a pond, two boys set stakes at A, B, and C. They measured the distance AC, 150 ft. and the distance BC, 175 ft. From C they sighted to A and B and drew lines in the direction of those two stakes. With a field protractor they measured ACB = 70°. They were now ready to make a drawing to scale. They first constructed an angle = 40 rd. B = C 70°. Letting 1" 50 ft., they made one side of the angle 3" long and the other side 3". They then measured the third side. It was 33" long. 3 X 50' = 187, the width of the pond. 1. Mr. Phelps wishes to measure the position of his barn. distance AB, but he cannot do it directly on account of the He takes the measurements shown on the drawing. Make a drawing to scale and find the distance from A to B. 75 ft. Barn 90 ft. B 2. Make a drawing of your school lot, using some convenient scale. Using the scale drawing, measure certain distances on the lot. 3. A square lot is 80 ft. on a side. Make a drawing of it and find the distance from one corner to the opposite corner. 4. Using your small protractor, see if you can make a large field protractor of cardboard. Have the base about 18 in. wide and mark every five degrees on it. 5. Using as tools the field protractor, some stakes, and a tape measure, take measurements to find some distances that would be difficult to measure directly. Draw triangles to scale and find the required distances. SQUARES AND SQUARE ROOT When a number is multiplied by itself, it is said to be squared. Thus, 3 squared is 3 × 3. The fact that a number is squared can be indicated by putting a small 2, called an exponent, to the right of and just above the given number. Thus 32 = 3 × 3 = 9, 52 5 X 5 = 25 X = If we wish to find the area of the square shown here, we multiply 4 by 4. This is why multiplying a number by itself is called "squaring" the number. The area of a square is found by squaring its side. The result is expressed in square units. A fraction is squared by squaring both its numerator and its denominator. 13. Find the areas of squares with these sides. 10 in. 16 ft. 14. A square lot is 75 ft. on a side. 25 yd. 18 in. 50 rd. Square Root We have seen that we can find the area of a square by squaring its side. It is also possible to reverse the process and find the side of a square when its area is known. The area of a square is 25 sq. in. How long is each side? We must find a number that multiplied by itself will give 25 as the product. The number is 5. The side of the square, then, is 5 in. 25 sq. in. The process of finding a number which when multiplied by itself will produce a given number, is called finding the square root of the given number. = Thus, the square root of 25 5, because 5 X 5 = 25. The square root of a number is one of its two equal factors. From the example given we see that the side of a square may be found by taking the square root of its area. Square root is indicated by this sign (√). Thus, √36, is read, "the square root of 36." The square root of a fraction may be found by taking the square root of the numerator and of the denominator of the fraction. Thus 9 25 = √9 = 3 √25 5 13. Find the lengths of the sides of squares with these areas. 81 sq. in. 100 sq. yd. 49 sq. ft. 121 sq. rd. 14. A square lot has an area of 64 sq. rd. of each side in rods. Find the length General Method of Finding Square Root So far we have considered only those numbers the square roots of which can be found by inspection. A more general method of finding the square root of a number is illustrated in these examples. A. Find the square root of 1,156. Beginning with the units figure, we divide the number into groups of two figures each, in this way: 1156. We next find the largest square that is contained in the left group, 11. This is 9. We place V1156 9 under 11, and subtract, and place the square root of 9, or 3, as the first figure of the required 3 9 2 root, directly over the first group of the given number. We The next step is to bring down the second group, 56. then double 3, the part of the root already obtained, and place the result, 6, at the left of 256. 3 The last step in this example is to find a 1156 number which can be placed with 6 to make the complete divisor and after 3 as the next figure of the root and as a multiplier of the complete divisor. This number is 4. 9 6 256 Placing 4 over the second group of the given number and multiplying 64 by 4, the second figure of the root, we get 256. Subtracting, the remainder is 0. |