EXERCISES-SIMILAR FIGURES

1. In the two similar triangles shown in § 115 suppose AB 1 foot, A'B' = 8 inches, and BC=1 feet. How long must B'C' be?

[HINT. Measure all lengths in inches and let x be the length of B'C'. Then by (a), § 115, we have 12/8 = 18/x.]

2. If a tree casts a shadow 50 feet long when a post 4 feet high casts a shadow 5 feet long, how high must the tree be?

3. A triangle has its sides 3 inches, 4 inches, and 5 inches long. Another triangle of the same shape has its shortest side 2 inches long. What are the lengths of the other sides of this triangle?

4. Of the two triangles in Ex. 3, the first has an area of 6 square inches. What is the area of the second?

[HINT. Let x be the area of the second. Then, by (b) of § 115, we have

5. Compare the areas of two city lots of the same shape if a side of the one is twice as long as the corresponding side of the other. Does your answer apply no matter what the shape is so long as it is the same for each lot?

6. What is the effect upon the area of a circle of trebling its diameter?

[HINT. Let D be the first diameter. Then 3D will be the second diameter.]

7. Compare the volume of a sphere whose diameter is 1 inch with that of a sphere whose diameter is 2 inches.

8. If a bottle of a certain shape holds 1 pint, how much will a similar bottle half as high hold?

9. A man whose eye is 5 ft. 6 in. above the ground sights over the top of a 12-foot pole and just sees the top of a tree. If he is 7 ft. from the pole and 63 ft. from the tree, how high is the tree?

[HINT. First draw a figure.]

10. The figure represents a kind of compasses used by draftsmen. By adjusting the screw at O, the lengths OA and OC, and the corresponding lengths OB and OD, may be changed proportionally. If OA=3 in. and OC-5 in., what part of the opening CD will the opening AB be?

B A

FIG. 57.

116. Mean Proportional.

If the means of a

proportion are equal, either mean is called the mean proportional between the extremes.

Thus, in the proportion 1:2

=

2:4, we have 2 as the mean proportional between 1 and 4. Likewise, in the proportion

6 is the mean proportional between 18 and 2, and in the proportion

we have r3 as the mean proportional between x1 and x2.

EXERCISES-MEAN PROPORTIONAL

1. Find the mean proportional between 6 and 24.

SOLUTION. Let x be the mean proportional. Then, 6/x = x/24. Whence, x2=144, and x=12. Ans.

Find the mean proportional between the two numbers given in each of the following exercises.

8. In the semicircle ABC, suppose a line CD drawn perpendicular to AB. Then (as shown in geometry) the length of CD will be a mean proportional

between the lengths AD and DB.

If AD=2 inches and DB=18 inches, find CD.

9. In Fig. 58, suppose AB=29 feet, and AD=4 feet. What is CD?

R

A D

B

FIG. 58.

10. The figure shows a circle and a point P outside it from which are drawn two lines PT and PS. The first of these lines just touches the circle and is called a tangent, while the second line cuts through the circle at two points R and S and is called a secant. In all such cases, the tangent PT is a mean proportional between the whole secant PS and its external part PR (as shown in geometry).

S

Ө

FIG. 59.

T

Find the length of PT if PR=93, and RS=50%.

CHAPTER XII

GRAPHICAL REPRESENTATION

117. Use of Diagrams. A few examples will show how diagrams are often used in everyday life to bring facts clearly before the eye.

EXAMPLE 1. A branch of the Y. M. C. A. wished to let people know of its progress in collecting money for a new

building. It placed a large signboard on the street and after ten days the board had the appearance shown in Fig. 60.

EXPLANATION. Two lines XX and YY had been drawn perpendicular to each other and each had been divided into equal units, beginning at the point where the lines cross. The points of division

were numbered 1, 2, 3, etc., as on a yardstick. Each unit on XX represented one of the days during which the money had been coming in, while each unit on YY represented $100. Starting at the point marked 1 on XX, the secretary of the Y. M. C. A. had drawn a heavy line extending upward until its end was on the level with the point marked 2 on YY. This indicated that on the first day just $200 was received. Similarly, he had drawn a heavy line beginning at the point marked 2 on XX and extending upward 2 units as measured on YY. This indicated that on the second day the amount received was $250. In the same way, he had drawn a heavy line upward corresponding to each of the 10 days. Read for yourself (from the scale on YY) the amount received on each day after the second.

EXAMPLE 2. Figure 61 shows the home expenses of a small family for a period of twelve months, beginning with January.

Each unit on XX represents 1 month, while each unit on YY represents $1. Read for yourself the amount spent during each of the twelve months.