Circle Theorems and Congruent Triangles Help!?

Question

A, B and C are three points on the circumference of a circle.

Angle ABC = Angle ACB.

PB and PC are tangents to the circle from the point P.

a) Prove that triangle APB and triangle APC are congruent.

b) Angle BPA = 10 degrees. Find the size of angle ABC.

Thanks!

Answer 1

Favorite Answer

(a)

In the triangles ABP, ACP:

AB = AC (sides of triangle ABC opposite equal angles ACB and ABC)

BP = CP (tangents from a point to a circle)

AP = AP (common)

Therefore triangles ABP, ACP are congruent [SSS].

(b)

Let O be the centre of the circle.

Join BO and CO.

As triangles ABP and ACP are congruent, angle CPA = 10 deg.

Angle BPC = angle BPA + angle CPA = 20 deg.

In quadrilateral OBPC:

Angle OBP = 90 deg (tangent perpendicular to radius)

Angle OCP = 90 deg (tangent perpendicular to radius)

Therefore OBPC is cyclic (opposite angles supplementary)

Anlge BOC = 180 deg - angle BPC

= 180 - 20

= 160 deg.

Angle BAC = angle BOC / 2

= 80 deg. (angle at the centre is twice that at the circumference)

In triangle ABC, therefore, as angle ABC = angle BCA:

Angle ABC = (1 / 2)(180 - angle BAC)

= (1 / 2)(180 - 80)

= 50 deg.