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