Circle Theorems and Congruent Triangles Help!?
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.
- Anonymous1 decade agoFavorite Answer
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].
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.