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.
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.
Given: ABC is an isosceles triangle, where AB = AC, circumscribing a circle.
To prove: The point of contact P bisects the base BC, i.e. BP = PC
Proof: It can be observe that
BP and BR; CP and CQ; AR and AQ; are pairs of tangents drawn to the circle
from the external points B, C, and A respectively.
Since the tangents drawn from an external point to a circle, then
BP = BR ............(i)
CP = CQ ............(ii)
AR = AQ ............(iii)
Given that AB = AC
AR + BR = AQ + CQ
BR = CQ .........[ From (iii) ]
BP = CP .........[From (i) and (ii) ]
P bisects BC.
OR
Given: The chord AB of the larger of the two concentric circles, with centre O,
touches the smaller circle ar C.
To prove: AC = CB
Construction: Let us join OC.
Proof: In the smaller circle, AB is a tangent to the circle at the point of contact C.
OC AB .....................(i)
( Using the property that the radius of a circle is perpendicular to the tangent at the point of contact )
For the larger circle, AB is a chord and from (i) we have OCAB
OC bisects AB
( Using the property that the perpendicular drawn from the centre to a chord of a circle bisects the chord.)
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.