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Diagonals of a Parallelogram from Coordinates

XAT 2015 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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Two diagonals of a parallelogram intersect each other at coordinates (17.5,23.5)(17.5, 23.5). Two adjacent points of the parallelogram are (5.5,7.5)(5.5, 7.5) and (13.5,16)(13.5, 16). Find the lengths of the diagonals.

  1. A.

    15 and 30

  2. B.

    15 and 40

  3. C.

    17 and 30

  4. D.

    17 and 40

  5. E.

    Multiple solutions are possible

Answer

D

Explanation

Let intersection point be X(17.5,23.5)X(17.5, 23.5) and vertices C(5.5,7.5)C(5.5, 7.5) and B(13.5,16)B(13.5, 16).

Distance CX=(17.5−5.5)2+(23.5−7.5)2=122+162=20CX = \sqrt{(17.5-5.5)^2 + (23.5-7.5)^2} = \sqrt{12^2 + 16^2} = 20. Length of diagonal AC=2×CX=40AC = 2 \times CX = 40.

Distance BX=(17.5−13.5)2+(23.5−16)2=42+7.52=16+56.25=72.25=8.5BX = \sqrt{(17.5-13.5)^2 + (23.5-16)^2} = \sqrt{4^2 + 7.5^2} = \sqrt{16 + 56.25} = \sqrt{72.25} = 8.5. Length of diagonal BD=2×BX=17BD = 2 \times BX = 17.

Lengths of diagonals are 17 and 40.

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