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Question
X and Y are points on the side LN of the triangle LMN such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z (See figure). Prove that ar (LZY) = ar (MZYX)
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Solution
Given: X and Y are points on the side LN such that LX = XY = YN and XZ || LM
To prove: ar (ΔLZY) = ar (MZYX)
Proof: Since, ΔXMZ and ΔXLZ are on the same base XZ and between the same parallel lines LM and XZ.
Then, ar (ΔXMZ) = ar (ΔXLZ) ...(i)
On adding ar (ΔXYZ) both sides of equation (i), we get
ar (ΔXMZ) + ar (ΔXXZ) = ar (ΔXLZ) + ar (ΔXYZ)
⇒ ar (MZYX) = ar (ΔLZY)
Hence proved.
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