ABCD is a parallelogram. If AB = 2AD and P is the mid-point of CD; prove that : ∠APB = 90°.
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ABCD is a parallelogram. If AB = 2AD and P is the mid-point of CD; prove that : ∠APB = 90°.

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Given: ABCD is a parallelogram. AB = 2AD and P is the mid point of CD. To prove: ∠APB = 90° Proof: Since ABCD is a parallelogram, the opposite sides of the parallelogram are equal. ⇒ AB = DC and AD = BC AB = 2AD ⇒ AD = AB and BC = AB ...............(1) Since P is the mid-point of CD and AB = DC, we have: DP = PC = DC = AB ...............(2) From equations (1) and (2), we conclude that AD = DP. In Δ APD, since two sides are equal, the angles opposite these sides are equal: ∠APD = ∠DAP ..................(3) Since, AB ∥ DC and AP is a transversal, the alternate interior angles are equal: ∠APD = ∠PAB ...............(4) From equation (3) and (4), we get: ⇒ ∠DAP = ∠PAB Let ∠DAP = ∠PAB = ∠A ...............(5) Similarly, from equations (1) and (2), BC = PC. In Δ BPC, since two sides are equal, the angles opposite to equal sides are equal: ∠CPB = ∠PBC ................(6) Since AB ∥ Dc and BP is a transversal, the alternate interior angles are equal: ∠CPB = ∠PBA ...............(7) From equations (6) and (7), we get: ⇒ ∠PBC = ∠PBA Let ∠PBC = ∠PBA = ∠B ...............(8) In a parallelogram, adjacent angles are supplementary: ∠A + ∠B = 180° ⇒ ∠A + ∠B = 180° ⇒ ∠A + ∠B = 90° In Δ APB, sum of all angles is 180°. ⇒ ∠APB + ∠ABP + ∠BAP = 180° ⇒ ∠APB + ∠B + ∠A = 180° ⇒ ∠APB + 90° = 180° ⇒ ∠APB = 180° - 90° ⇒ ∠APB = 90° Hence, ∠APB = 90°

Mathematics

ABCD is a parallelogram. If AB = 2AD and P is the mid-point of CD; prove that : ∠APB = 90°.

Rectilinear Figures

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Mathematics

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