Mathematics — Standard (041) — CBSE Class 10 board question

(c) 6·5 cm

Using the property that tangents from an external point are equal: AP = BP (from point A on chord AC... )

Let AP = x, then CP = AC − x = 7 − x. Since tangents from A: AB = 10, and tangents from C: CB = 10.
BP = BC − CP = 10 − (7 − x). Also BP = AB − AP = 10 − x.
So: 10 − (7 − x) = 10 − x → 3 + x = 10 − x → x = 3·5.
∴ BP = 10 − 3·5 = 6·5 cm.

• Key theorem: tangents from an external point to a circle are equal in length.
• Here, A and C are points on the circle; P is the point where line BC is tangent to the circle. Treat A and C as external points with their respective tangent lengths to find AP and CP, then subtract from BC = 10 cm.
• Answer is (c) 6·5 cm.