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ANGLES AND SUBTENDED ARCS

It is proved in geometry that angles at the center of a circle are proportional to the arcs which they subtend. Angle : angle y = arc ab:

a

SOLUTION.

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Example. The radius of a circle is 15 in. Find the length of an arc 37° 30' of this circle.

Circumferenceπ × diameter.

... circumference = 3.1416 x 30 in.=94.248 in. .. Arc ab: 94.248 = 37.5: 360.

.. 360 x arc ab

94.248 × 37.5.

Arc ab

9.82. Ans. 9.82 in.

EXERCISE 158

1. The radius of a circle is 37 in. Find the length of an arc of 72° of this circle.

2. The radius of a circle is 94 in. Find the length of an arc of 30° of this circle.

3. What is the length of an arc of 1° on a circle whose radius is 58 ft.?

4. What angle does an arc of 40.212 ft. subtend at the center of a circle whose radius is 64 ft.?

5. What angle does an arc of 6.032 ft. subtend at the center of a circle whose radius is 96 ft.?

6. The distance of the moon from the earth is 239,000 mi., and the diameter of the moon is 2170 mi. To an observer on the earth, what angle does the moon's diameter subtend?

Using for a measure the side of a small square as ruled, how long is CB? CF? CH? AB? EF? GH?

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III

Is this a true proportion, — AB: EF = CB: CF?
Is this a true proportion,- EF: GH CF: CH?
What kind of figures are ABC, EFC,
Name their bases. Their altitudes.

GHC?

What is the length of the base of I? of II? of III? What is the length of the altitude of I? of II? of III? Are the following proportions true?

Base of I base of II = altitude of I: altitude of II;

base of I base of III altitude of I: altitude of III;

:

base of II: base

III.

=

of

III =

= altitude of II: altitude of

SIMILAR FIGURES

Similar figures are figures having the same form. Triangles ABC, EFC, and GHC are similar triangles. Parallelograms I, II, and III are similar parallelograms. All regular polygons of the same number of sides are similar figures. The drawing which a surveyor makes of a tract of land is similar to the tract of land.

It is shown in geometry that corresponding dimensions of similar figures have the same ratio.

We noticed this to be true in the above figures. This fact is useful in computation.

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1. When a tree 90 ft. high casts a shadow 75 ft. long, find the length of the shadow cast by a pole 24 ft. high.

2. How high is an object which casts a shadow 110 ft. when a pole 8 ft. high casts a shadow 5 ft.?

3. A map is drawn to a scale of 40 mi. to 1 in. On this map two cities are 2 in. apart. How many miles are there between these cities?

4. In a map of a city two public buildings are 91 in. distant. If the map is drawn to the scale of 1 in. to of a mile, how far is it from one of these buildings to the other? Refer again to the illustrations.

Compute the area of triangle ABC. Of EFC. Of GHC. Of parallelogram I. Of II. Of III.

Find the values of the following ratios. Area ABC;

area EFC; area ABC: area GHC; area EFC: area GHC; area I area II; area I: area III; area II: area III.

Are the following proportions true?

Area ABC: area EFC = AB2 : EF2.
Area EFC: area GHC = FC2: HC2.

Area I: area II = square of base I: square of base II.
Area II area III
:
square of altitude II: square of

altitude III.

=

The areas of similar figures are to each other as the squares of their corresponding dimensions.

This fact is also useful in computations.

Example. The area of a triangle, one of whose sides is 5 rd., is 11 sq. rd. Find the corresponding side of a similar triangle whose area is three times as great.

SOLUTION. Let X equal the required side.

The area of the first triangle: the area of the second triangle square of the side of the first triangle: square of the side of the second triangle.

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X= √75 = 8.662. Ans. 8.662 rd.

5. The area of a triangle is 15 sq. ft., and one of its sides is 10 ft. Find the corresponding side of a similar triangle five times as large.

6. The altitude of a triangle is 10 ft. If the triangle is divided into two equal parts by a line parallel to its base, how far from the vertex must this line be drawn?

7. Corresponding sides of two similar quadrilaterals are in the ratio of 4 to 11. Find the ratio of their areas.

8. The diameters of two circles are 12 and 18 in. Find the ratio of their areas. Find the ratio of their radii. Find the ratio of their circumferences.

9. The distance between two cities is 90 mi., and on a map containing both cities their positions are distant 55 in. What area is represented by a circle of in. radius on this map?

SURFACES OF THE PRISM, PYRAMID, CYLINDER,

CONE, AND SPHERE

A right prism is a solid, two of whose faces are equal and parallel polygons, and whose other faces are rectangles. The upper and lower faces are called the bases, and the other faces are called lateral faces.

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A prism is named according to the shape of its bases. Thus, a prism whose bases are triangles is called a triangular prism. The area of the surface of a prism can be computed by adding together the areas of the separate faces.

The following rule may easily be established experimentally by paper cutting or other devices:

The lateral surface of a right prism equals the product of the perimeter of its base by the height of the prism.

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