In the picture below, the radius of the circle is 15 centimetres. Compute the lengths of the tangents PQ and PR

(i) The point P is equidistant from A and B

(ii) The line OP bisects the line AB and the angle APB

Open in App

Solution

Given: OR = OQ = 15 cm and ∠ROQ = 120°

Construction: Join OP.

In ΔOPR and ΔOPQ:

OR = OQ (Radii of the same circle)

OP = OP (Common side)

PR = PQ (Length of tangents drawn from a point outside the circle are equal)

If all the sides of one triangle are equal to the corresponding sides of the other triangle then the two triangles are congruent.

∴ ΔOPR ≅ ΔOPQ

⇒ ∠POR = ∠POQ (Congruent parts of congruent triangles are congruent)

⇒ ∠POR = ∠POQ =

In ΔPOR:

= tan 60°

⇒ RP = OR tan 60°

= 15 cm ×

= 15 × 1.732 cm

= 25.98 cm

We know that lengths of tangents drawn from an external point are equal.

∴ PQ = PR = 25.98 cm

Suggest Corrections

Similar questions

In the circle centred at O, the tangents at A and B intersect at P. Prove the following:

(i)

the point P is equidistant from A and B

(ii)

the line OP bisects the line AB and the angle APB

(iii)

if the line OP cuts the line AB at Q, then OQ × OP = r², where r is the radius of the circle

Q. P is a point equidistant from two lines l and m intersecting at point A. Show that the line AP bisects the angle between them.

The line joining the centre of a circle and the point from which two tangents are drawn bisects the angle between the tangents.

In the given figure, two circles touch each other externally at point $P$ . $A B$ is the direct common tangent of these circles. Prove that :

(i) tangent at point $P$ bisects $A B$ ,

(ii) angle $A P B =$ $90^{\circ}$ .

PQ and RQ are chords of a circle equidistant from the centre and S is a point on the circle,such that QS forms diameter. Prove that the diameter QS bisects $∠$ PQR and $∠$ PSR.

Join BYJU'S Learning Program

In the picture below, the radius of the circle is 15 centimetres. Compute the lengths of the tangents PQ and PR (i) The point P is equidistant from A and B (ii) The line OP bisects the line AB and the angle APB

In the picture below, the radius of the circle is 15 centimetres. Compute the lengths of the tangents PQ and PR

(i) The point P is equidistant from A and B

(ii) The line OP bisects the line AB and the angle APB