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Through the m...

Question

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting diagonal AC in L and AD produced in E. Prove that : EL = 2 BL.

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Solution

1 = 6 (Alternate interior angles)

2 = 3 (Vertically opposite angles)

DM = MC (M is the mid-point of CD)

So, DE = BC (Corresponding parts of congruent triangles)

Also, AD = BC (Opposite sides of a parallelogram)

AE = AD + DE = 2BC

Now, 1 = 6 and 4 = 5

$increment E L A tilde increment B L C space space space open parentheses A A space s i m i l a r i t y close parentheses fraction numerator E L over denominator B L end fraction equals fraction numerator E A over denominator B C end fraction fraction numerator E L over denominator B L end fraction equals fraction numerator 2 B C over denominator B C end fraction equals 2 E L equals 2 B L$

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