CIRCLES (revision)

CIRCLES (revision)

1. If a hexagon ABCDEF  circumscribe  a circle ,prove that AB+CD+EF=BC+DE+FA.

2. Let “s” denote the semi-perimeter of a ∆ABC in which BC=a ,CA=b , AB=c. If a circle touches the sides BC, CA, AB at D, E, F, respectively  ,  prove that BD=s-b.

3. From an external point P, two tangents ,PA and  PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If  PA=10 cm , find the perimeter of triangle PCD.

4. ABC is an isosceles triangle , in which AB=AC , circumscribed about a circle. Show that BC is bisected at the point of contact .

5. Two tangents PA and PB from a P to a circle with centre O are inclined to each other at an angle of 80⁰ ,then find ∟POA.

6. In two concentric circles  , a chord of length 24m of larger circles becomes a tangent to the smaller circle whose radius is 5 cm . Find the radius of the larger circle.

7. PQR is a right triangle right angled at Q  . PQ=5cm, QR=12cm .A circle with centre O is inscribed in ∆PQR, touching  its all sides . Find the radius of the circle  .  

8. AB is a chord of length 24cm of a circle of radius 13cm. The tangent at A and B intersects at a point C. Find the length of AC.  

9. P  is the mid point of an arc  QPR of a circle .Show that the tangent at P is parallel to the chord QR.

10. Prove that parallelogram circumscribing a circle is a rhombus.