In Fig. 5, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents drawn at P and Q intersect at T. Find the length of TP.

(Fig. 5)

Question

In Fig. 5, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents drawn at P and Q intersect at T. Find the length of TP.(Fig. 5)

Philoid · Accepted Answer

PT is perpendicular to OP (Tangents are always perpendicular to the center)

PQ is perpendicular to OT (The line joining the Point of contact of common tangents with the center is perpendicular to the common tangent)

Let length of PT be x and length of RT be y

Since PR is perpendicular to OT so by using Pythagoras theorem we calculate OR

Length of PR = 4 cm (Half of PQ)

OR² = 5² - 4²

⇒ OR = 3 cm

Using Pythagoras theorem:

PR² + RT² = PT²

⇒ 16 + y² = x² … Equation(i)

PT² + OP² = OT²

⇒ x² + 5² = (3 + y)²

⇒ x² + 5² = 9 + y² + 6y

Putting the value of x² from equation (i) we get

16 + y² + 5² = 9 + y² + 6y

⇒ 6y = 32

⇒ y =

Putting the above value in Equation (i) we get

⇒

Answer: The length of PT =