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Equal Angles

Problem

In the square ABCD, M is the midpoint of AB.
A line is drawn through M perpendicular to CM to locate N.

Prove that the size of angle BCM is equal to the size of angle MCN.


Solution

We begin by drawing a line through M and parallel to BC to produce E on CN and F on CD.

As right angle triangle CMN is half of a rectangle, CN is one diagonal, and because E is the midpoint of CN, CE = EN = EM.

In which case, triangle CEM is isosceles, a = b (base angles equal).

Because MF is parallel to BC, a = c (alternate angles).

Hence b = c, and we prove that the size of angle BCM is equal to the size of angle MCN.

If AB=4, find the perimeter of triangle CDN.

Problem ID: 196 (21 Dec 2004)     Difficulty: 2 Star

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