Tangents PA and PB are drawn from an external point P to two concentric circle with centre O and radii 8 cm and 5 cm respectively, as shown in Fig., If AP = 15 cm, then find the length of BP.
Given: Tangents PA and PB are drwan from an external point P to two
concentric circles with centre O and radii OA = 8 cm, OB = 5 cm
respectively. Also, AP = 15 cm.
Construction: We join the points O and P.
Proof: OA AP ; OB BP
[ Using the property that radius is perpendicular to the tangent at the
point of contact of a circle.]
In right angled triangle OAP,
OP2 = OA2 + AP2 [ Using pythagoras theorem ]
= (8)2 + ( 15 )2 = 64 + 225 = 289
OP = 17 cm
In right angled triangled OBP,
OP2 = OB2 + BP2
BP2 = OP2 - OB2
(17)2 - (5)2
289 - 25
= 264
BP = = 2 cm.
In fig., an isosceles triangle ABC, with AB =AB, circumscribes a circle. Prove that the point of contact P bisects the base BC.
OR
In fig., the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.
In fig., OABC is a square of side 7 cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region.
A point P divides the line segment joining the points A(3,-5) and B(-4, 8) such that . If P lies on the line x + y = 0, then find the value of K.
If the vertices of a triangle are (1, -3), (4, p) and (-9, 7) and its area is 15 sq. units, find the value (s) of p.
Prove that the parallelogram circumscribing a circle is a rhombus.
OR
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid.
OR
A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.