Mathematics
Two points A and B lie on the same side of a line XY. If AD ⊥ XY and BE ⊥ XY meet XY in D and E respectively and C is the mid-point of AB, show that CD = CE.
Mid-point Theorem
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Answer
Join BD which intersects CF at O.
Given,
CF ⊥ XY, AD ⊥ XY and BE ⊥ XY
⇒ CF || AD || BE
By mid-point theorem,
The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it.
By converse of mid-point theorem,
A line drawn through the midpoint of one side of a triangle, and parallel to another side, will bisect the third side.
Since,
⇒ CF || AD
⇒ CO || AD
In △ADB,
C is the mid-point of AB and CO || AD.
∴ O is the mid-point of BD. (By converse of mid-point theorem)
Given,
⇒ CF || BE
⇒ OF || BE
In △BDE,
O is the mid-point of BD and OF || BE.
∴ F is the mid-point of DE. (By converse of mid-point theorem)
⇒ DF = FE
In △CDF and △CEF,
⇒ DF = FE (Proved above)
⇒ CF = CF (Common side)
⇒ ∠CFD = ∠CFE (Both equal to 90°)
∴ △CDF ≅ △CEF (By S.A.S. axiom)
⇒ CD = CE (Corresponding parts of congruent triangles are equal)
Hence, proved that CD = CE.
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