Midpoint Theorem on Right-angled Triangle
Here we will prove that in a right-angled triangle the median drawn to the hypotenuse is half the hypotenuse in length.
Solution:
Given: In ∆PQR, ∠Q = 90°. QD is the median drawn to hypotenuse PR.
To prove: QS = \(\frac{1}{2}\)PR.
Construction: Draw ST ∥ QR such that ST cuts PQ at T.
Proof:
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Statement |
Reason |
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1. In ∆PQR, PS = \(\frac{1}{2}\)PR. |
1. S is the midpoint of PR. |
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2. In ∆PQR, (i) S is the midpoint of PR (ii) ST ∥ QR |
2. (i) Given. (ii) By construction. |
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3. Therefore, T is the midpoint of PQ. |
3. By converse of the Midpoint Theorem. |
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4. TS ⊥ PQ. |
4. TS ∥ QR and QR ⊥ PQ |
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5. In ∆PTS and ∆QTS , (i) PT = TQ (ii) TS = TS (iii) ∠PTS = ∠QTS = 90°. |
5. (i) From the statement 3. (ii) Common side. (iii) From the statement 4. |
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6. Therefore, ∆PTS ≅ ∆QTS. |
6. By SAS criterion of congruency. |
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7. PS = QS. |
7. CPCTC |
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8. Therefore, QS = \(\frac{1}{2}\)PR. |
8. Using statement 7 in statement 1. |
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