Question

In the given figure, PT is a tangent to the circle at T, chord BA is produced to meet the tangent at P. Perpendicular BC bisects the chord TA at C. If PA = 9 cm and TB = 7 cm, find the lengths of:

1. AB
2. PT

Answer 1

given:

• PA = 9 cm

• TB = 7 cm

Step 1: Find AB

PA = 9 cm, and AB = AP + PB

From the figure:

TB = 7cm

TB is part of line PB, and the segment AB = AP + PB

Since TB is part of the circle and AB is the chord, and point T is on the circle, while PA is extended, we observe that:

PB = PA + AB ⇒ But from figure, actually: AB = PB − PA

AB = AP + PB = 9 + 7 = 16 cm​

Step 2: Use right-angled triangle △TBC to find PT

BC ⊥ TA, and it bisects TA, so:

TC = CA

TB = 7 cm

Let’s consider triangle △TBC, which is right-angled at C.

Use Pythagoras Theorem:

`CB = sqrt(TB^2-TC^2) = sqrt(7^2-x^2)`

Use tangent-secant theorem:

(PT)² = PA⋅PB

We are given:

PA = 9 cm

PB = PA + AB = 9 + 7 = 16 cm

PT² = PA⋅PB = 9⋅16 = 144 ⇒ PT = `sqrt144` = 12 cm

Final Answers:

AB = 16 cm
PT = 12 cm