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Section IV.-CIRCLES, TANGENTS,

AND ARCS.

DEFINITIONS.

1. The area of a figure is its superficies or surface. Such measure

ments are calculated by square or superficial measure. Thus
(a) A square whose side is 4 linear inches, contains an area of

· 16 square inches. Ex. AB

(6) A rectangle whose adjacent sides are 6 and 3 linear feet,

contains an area of 18 square feet. Ex. AB

BIC

2. Concentric circles. Circles are said to be concentric when they

have a common centre. Ex. A, B, C

Problem 45. To find the centre of a given circle A.

1. Draw any chord BC, and bisect it by a line meeting the

circumference in D and E (Pr. 1).

2. Bisect ED by the line FG. The point of intersection, A,

is the centre of the given circle.

Problem 46. To describe a circle which shall pass through three given points A, B, and C.

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1. Join AB, and bisect it by the perpendicular DE (Pr. 1.)

2. Join BC, and bisect it by the perpendicular FE, inter

secting DE in E. 3. With centre E, and radius EA, describe the required

circle, and it will pass through the three given points A, B, C.

Problem 47. To describe a circle which shall pass through any given point A, and which shall also be tangential to the given line BC, in a given point D.

1. Draw DE at right angles to BC (Pr. 2).

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2. Join AD, and make the angle DAF equal to the angle

ADE (Pr. 10). 3. Then with centre F, and radius FD, describe the

required circle, which will pass through the given point A, and be tangential to the given line BC, at the given

point D.

Problem 48. To divide the area of a given circle A into any number of equal parts by concentric circles (say three in this case).

1. Draw any radius A B, and divide it into three equal parts

in the points C, D (Pr. 15).

2. On AB describe a semi-circle, and from the points of

division C and D, erect perpendiculars to AB (Pr. 2). meeting the semicircle in E and F.

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3. From A as centre, with AE and AF as radii, describe

circles. Then the areas 1, 2, 3, contained between these circles, will be equal.

Problem 49. To divide a given circle into any number of parts, which shall be equal both in area and outline.

1. Draw any diameter AB, and divide it into the required

number of parts (say four) (Pr. 15) in the points 1, 2, 3.

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2. Bisect A1, and describe a semicircle on it, and a similar

one below 3B.

3. Bisect 1B, the remaining part of the line AB, and

describe a semicircle below the line, and with the same

radius a similar one on the line A3. 4. With points 1 and 3 as centres, describe semicircles on

the line A 2 and below 2 B respectively. Then the given circle will be divided into the required four parts which are equal both in area and outline.

Problem 50. To describe a circle touching two given circles A and B, and one of them in a given point C.

1. Join the centres of the two circles A and B by the

straight line AB. 2. Draw from C, the given point of contact, a radius, CB.

3. In the other circle, draw a radius AD, parallel to BC

(Pr 8). 4. Join CD, producing it if necessary to a point opposite to

C, as E. 5. Join CB and EA, and produce them until they meet in

F. Then FC or FE will be the radius of the required circumscribing circle.

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