# From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD

**Solution:**

Given, PA and PB are the two __tangents__ drawn to a circle with centre O though an external point P.

At point E on the circle the tangent is drawn which intersects PA and PB at C and D.

Given, PA = 10 cm

We have to find the __perimeter of the triangle__ PCD.

We know that the tangents drawn to a circle through an external point are equal.

From the figure,

The tangents drawn through point P are PA and PB

So, PA = PB -------------- (1)

The tangents to the circle through C are CA and CE

So, CA = CE -------------- (2)

The tangents to the circle through D are DB = DE

So, DB = DE -------------- (3)

Considering triangle PCD,

__Perimeter__ = PC + PD + CD

From the figure,

CD = CE + DE

So, perimeter = PC + PD + CE + DE

From (2) and (3),

Perimeter = PC + PD + CA + DB

On rearranging,

Perimeter = PC + CA + PD + DB

From the figure,

PC + CA = PA

PD + DB = PB

By (1), PA = PB

So, perimeter = PA + PB

Perimeter = PA + PA

= 2PA

Given, PA = 10 cm

So, 2PA = 2(10)

= 20 cm

Therefore, the perimeter of the triangle PCD is 20 cm.

**✦ Try This:** If tangents PA and PB from a point P to a circle with center O are drawn so that ∠APB = 80°, then, ∠POA?

**☛ Also Check:** NCERT Solutions for Class 10 Maths Chapter 10

**NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 3**

## From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD

**Summary:**

From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, the perimeter of the triangle PCD is 20 cm

**☛ Related Questions:**

- If AB is a chord of a circle with centre O, AOC is a diameter and AT is the tangent at A as shown in . . . .
- Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and . . . .
- In a right triangle ABC in which ∠B = 90°, a circle is drawn with AB as diameter intersecting the hy . . . .

visual curriculum