If r1and r2are the radii of the two non-intersecting non-enclosing circles, Length of the direct common tangent =2212)r-(r-centre)between(DistanceLength of transverse common tangent = 2212)r(r-centre)between(Distance+Two circles are said to be concentric if they have the same centre. As is obvious, here the circle with smaller radius lies completely within the circle with bigger radius. Arcs and Sectors Fig. 4.37 An arc is a segment of a circle. In Fig. 4.37, ACB is called minor arc and ADB is called major arc. In general, if we talk of an arc AB, we refer to the minor arc. AOB is called the angle formed by the arc AB (at the centre of the circle). The angle subtended by an arc at the centre is double the angle subtended by the arc in the remaining part of the circle. In Fig. 4.37, ∠AOB = 2 . ∠AXB. = 2.∠AYB Angles in the same segment are equal. In Fig. 4.37, ∠AXB = ∠AYB. Fig. 4.38 The angle between a tangent and a chord through the point of contact of the tangent is equal to the angle made by the chord in the alternate segment (i.e., segment of the circle on the side other than the side of location of the angle between the tangent and the chord). This is normally referred to as the "alternate segment theorem." In Fig. 4.38, PQ is a tangent to the circle at the point T and TS is a chord drawn at the point of contact. Considering ∠PTS which is the angle between the tangent and the chord, the angle TRS is the angle in the "alternate segment". So, ∠PTS = ∠TRS. Similarly, ∠QTS = ∠TUS. Fig. 4.39 We have already seen in quadrilaterals, the opposite angles of a cyclic quadrilateral are supplementary and that the external angle of a cyclic quadrilateral is equal to the interior opposite angle. The angle in a semicircle (or the angle the diameter subtends in a semicircle) is a right angle. The converse of the above is also true and is very useful in a number of cases - in a right angled triangle, a semi-circle with the hypotenuse as the diameter can be drawn passing through the third vertex (Refer to Fig. 4.39). R S T Q P U X Y A B •O D C

Triumphant Institute of Management Education Pvt. Ltd. (T.I.M.E.) HO:95B, 2ndFloor, Siddamsetty Complex, Secunderabad – 500 003.Tel : 040–27898195 Fax : 040–27847334 email : [email protected]website : SM1001908/51 B A o θFig. 4.40 The area formed by an arc and the two radii at the two end points of the arc is called sector. In Fig. 4.40, the shaded figure AOB is called the minor sector. AREAS OF PLANE FIGURESMensuration is the branch of geometry that deals with the measurement of length, area and volume. We have looked at properties of plane figures till now. Here, in addition to areas of plane figures, we will also look at surface areas and volumes of "solids." Solids are objects, which have three dimensions (plane figures have only two dimensions).

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