5. Through a point O within a circle, two straight lines A B, A'B' are drawn to the circumference; A B=18 in., A O=12, A'O-3 in. Find A'B'. Ans. 27 in.

6. If in the above case A B=31 in., A'B'=32 in., and O B= 15, find the length of the segments of A'B'. Ans. 20 and 12 in. 7. Two chords A B, A' B' when produced meet in a point without the circle. The rectangle contained by O A, O B is 63 square feet, find the area of the rectangle whose sides are respectively equal to O A', A'B' the sides Ŏ A being 3 ft. Ans. 54 square ft.

8. A BB'A' is a quadrilateral inscribed in a circle; A and B' are points in the same diameter. If A B, A'B' meet in O and AB=43, A'B'=7}, OA=5, OB=4 in.; find the area of the quadrilateral. Ans. 53 square in.

CHAPTER IX.

TANGENTS.

216. A tangent to a circle is a straight line which has one point in common with the circumference, and only The tangent to a circle has many important properties. Thus :

one.

A

M

T

Every straight line perpendicular to the extremity of a diameter is a tangent to the circumference. 217. Let A T be a perpendicular to the extremity A of the diameter (fig. 193); join with the centre any point M of this perpendicular; the line O M is an oblique line with respect to the straight line AT, wherefore it is longer than the perpendicular O A, which is the radius of the circle. Therefore M is outside

Hence the point A is the only point of the line AT in the circumference; hence it follows from the definition that this straight line is a tangent.

The point A is called the point of contact. following problems naturally arise in this place.

To draw a tangent.

The

1st. From a given point in the circumference. 218. It is sufficient to erect a perpendicular at the extreme of the radius.

2nd. Through a given point without the circle. Join the given point A to the centre O (fig. 194), and upon this line O A, as diameter, describe a circle. It cuts the given circle in two points,

D and D', which will be the points of contact of the tangents required; hence, by joining the points AD, A D', we shall have two tangents through the point A, for the angles A DO, A D'O, inscribed in a semicircumference, are right-angles.

Fig.194.

We might, however, draw the tangent without sensible error by applying a ruler to the point A, and moving it so that it touches the circle. This method may be as exact practically as the first, and even more so, for the second construction, although theoretically correct, requires various operations, the accuracy of which depends upon the skill of the operator and the perfection of his instruments.

Thus error may be made in determining the middle point of A O, for example, for there are three operations to find this point, and each is liable to error; precision may also be wanting in describing the circle. upon A O as diameter.

The figure shows that the diameter A O is a sym. metrical axis. Therefore,

The line bisecting the angle formed by two concurrent tangents passes through the centre.

219. It will be seen directly that since every point in the line bisecting the angle DA D' is equidistant from the sides of the angle, the point O, equidistant from D and D', lies in the bisecting line.

This line of bisection is perpendicular to the middle point of the chord of contact, and the parts of the tangents intercepted between their point of intersection and the points of contact are of the same length.

220. If the figure be turned round O A, the tangent AD will fall on A D', since the perpendiculars drawn from the point O upon these lines will coincide; thus, the tangents drawn from the point A to the circle are equal.

Draw D D', the chord of contact, the line O A passing through the two points O and A, which are each equidistant from the points D and D', will be perpendicular to D D' at its middle point.

To draw a tangent common to two given circles. 221. Let O and I be the two circles (fig. 195). Take with the compasses a length equal to the difference between the radii of the two circles, and describe a circumference from the centre I of the greater circle. Upon OI as diameter, describe a circle which shall cut the preceding one in two points A and A'; draw the radii I A, IA', and produce them to B B', and lastly through the points B and B', draw parallels to the straight lines A O, A O'. These lines will be the two common tangents called exterior tangents.

In order to draw the two interior tangents, it is necessary to make the same series of constructions,

only the circumference from the centre I must be described with a radius equal to the sum of the radii of the two circles (fig. 196).

The same problem may be solved, basing our work upon the principle demonstrated (§ 213); for a tangent common to two circles may be considered as a secant joining the extremities of two parallel radii, which are at the same time perpendicular to this secant; so that the tangents common to two circles pass through the fixed points C and C' determined as in § 213 (fig. 197). If then we draw through the points C and C' tangents to one of the circles, they will at the same time be tangents to the other circle.

When the two circles are equal.

222. This process is not applicable when the radii of the two circles are nearly equal, because the point C will be then very far distant, and the circle described from the centre O will be very small, since its radius is

equal to the difference of the radii of the two circles. In this case it is easier to draw the tangent by applying a ruler to the circles, for although

ex

this method rests upon no theoretical consideration, it is in this case less liable to error than those taught by theory. When the radii of the two circles are equal, the common terior tangents are parallel to the line joining their centres, for these lines joining the extremities of equal and parallel straight lines, are themselves equal and parallel.

The processes of § 221 are both inapplicable in this case (fig. 198). We may proceed thus: drawthrough the centre of one of the circles a diameter perpendicular to the line joining the centres; also draw parallels to this line at the extremity of the diameter; these parallels will be the two common exterior tangents.

Fig. 198.

The methods of § 221 are available for the construction of interior tangents. The point C' is then the middle point of the line between the centres O and I.

To draw a tangent to a given circle parallel to a given straight line.

223. From the centre let fall a perpendicular to the given straight line (fig. 199). This perpendicular will meet the circumference in the points A and B. Through these points draw parallels to the given straight line, and they will be the tangents required.

To draw a tangent which shall make with a given straight line an angle equal to a given angle. 224. Let AB (fig.200) be the given line. At any point