? asked in Science & MathematicsMathematics · 1 decade ago

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.

Thanks!

1 Answer

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  • Anonymous
    1 decade ago
    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.

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