Question

(i) 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 = 9cm and TB = 7cm, find the lengths of: [2] (a) AB (b) PT​

Answer 1

Given:

PA=9cm

TB=7cm

To find:

The length of AB and PT

Solution:

The length of AB is 7 cm and of PT is 12 cm.

We can find the lengths by following the given steps-

We know that the perpendicular BC divides TA into two equal parts.

So, TC=CA.

Now, in ΔTCB and ΔACB,

TC=CA (given)

Angle TCB=Angle ACB=90° (given)

CB is the common side.

So, ΔTCB and ΔACB are congruent using the SAS rule.

Since the two triangles are congruent, their corresponding sides will also be equal.

So,  TB=AB=7cm.

Now, we know that PT is a tangent to the circle and PAb is a secant.

PA×PB=$PT^{2}$

PA=9cm

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

On putting the given values, we get

9×16=$PT^{2}$

3×4=PT

PT=12cm

Therefore, the length of AB is 7cm and of PT is 12 cm.

Answer 2

Answer:

7cm and 12 cm

Step-by-step explanation:

The length of AB is 7 cm and of PT is 12 cm.

We can find the lengths by following the given steps-

We know that the perpendicular BC divides TA into two equal parts.

So, TC=CA.

Now, in ΔTCB and ΔACB,

TC=CA (given)

Angle TCB=Angle ACB=90° (given)

CB is the common side.

So, ΔTCB and ΔACB are congruent using the SAS rule.

Since the two triangles are congruent, their corresponding sides will also be equal.

So,  TB=AB=7cm.

Now, we know that PT is a tangent to the circle and PAb is a secant.

PA×PB=

PA=9cm

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

On putting the given values, we get

9×16=

3×4=PT

PT=12cm

Therefore, the length of AB is 7cm and of PT is 12 cm.