Proof By the Equal Intercepts Theorem

Here we will prove that in the given XYZ, M and N are the midpoints of XY and XZ respectively. T is any point on the base YZ. Prove that MN bisects XT.

Solution:

Given: In ∆XYZ, XM = MY and XN = NZ. MN cuts XT at U.

To prove: XU = UT.

Construction: Through X, draw PQ ∥ YZ.

Proof:

 Statement Reason 1. MN ∥ YZ. 1. The line segment joining the midpoints of the two sides of a triangle is parallel to the third side. 2. PQ ∥ MN ∥ YZ. 2. PQ ∥ YZ and MN ∥ YZ. 3. The transversal XY makes equal intercepts on PQ, MN and YZ. 3. Given that XM = MY. 4. The transversal XT also makes equal intercepts on PQ, MN and YZ. 4. By the Equal Intercepts Theorem. 5. XU = UT. (Proved) 5. From statement 4.

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