15. In AABC, ZB = 90° and D is the mid-point of BC. Prove that :
(i) AC2 = ADP + 3CD2
(ii) BC2 = 4(AD2 - AB2)
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Answer:
Given: In △ABC, ∠B = 90° and D is the mid-point of BC.
To Prove: AC² = AD² + 3CD²
Proof:
In △ABD,
AD² = AB² + BD²
AB² = AD² - BD².......(i)
In △ABC,
AC² = AB²+ BC²
AB² = AC²- BD² ........(ii)
Equating (i) and (ii)
AD² - BD² = AC² - BC²
AD²- BD²= AC² - (BD + DC)²
AD² - BD²= AC² - BD²- DC²- 2BDx DC
AD² = AC² - DC² - 2DC² (DC = BD)
AD²= AC² - 3DC²
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