Question

PQ is a chord of length 16 cm of a circle of radius 10 cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP.

Answer 1

Given:

PQ = 16 cm, circle radius OP = OQ = 10 cm.

Tangents at P and Q meet at T. (Need TP)

Step-wise calculation:

1. Let R be the midpoint of chord PQ.

Then PR = RQ

= `16/2`

= 8 cm

2. OR is perpendicular to PQ and bisects it, so `OR = sqrt(OP^2 - PR^2)`

= `sqrt(10^2 - 8^2)`

= `sqrt(100 - 64)`

= 6 cm

3. Note OT passes through R and TR = OT – OR.

Triangles TRP and RPO are similar right triangles with a common acute angle, hence `(TP)/(PO) = (RP)/(RO)`.

4. Substitute the known values:

`(TP)/10 = 8/6` 

⇒ `TP = 10 × 8/6`

= `80/6`

= `40/3` cm 

TP = `40/3` cm (≈ 13.333... cm).