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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

In a trapezoid A B C D , the internal bisector of angle A...

In a trapezoid

A B C D

, the internal bisector of angle

A

intersects the base

B C

(or its extension) at the point

E

. Inscribed in the triangle

A B E

is a circle touching the side

A B

at

M

and side

B E

at the point

P

. Find the angle

D A E

in degrees, if

A B: M P=2

.

In a trapezoid A B C D , the internal bisector of angle A intersects the base B C (or its extension) at the point E .

Correct answer

60

Step-by-step solution

In a trapezoid A B C D , the internal bisector of angle A intersects the base B C (or its extension) at the point E .

A D \| B C

Given,

P M=\frac{x+y}{2}

\begin{aligned} & \angle B A E=\angle A E B=\theta \\ \Rightarrow \quad & x=z \end{aligned}

By symmetry

M

is mid-point of

B A

\begin{array}{ll} \therefore & \frac{x+y}{2}=y \Rightarrow x=y \\ \therefore & \angle B=60^{\circ} \Rightarrow \angle A=60^{\circ} \end{array}

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