Area of triangle given perpendicular medians

CAT 2019 Slot 2 · QA · Medium · Geometry

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In a triangle ABCABC, medians ADAD and BEBE are perpendicular to each other, and have lengths 12 cm12\text{ cm} and 9 cm9\text{ cm}, respectively. Then, the area of triangle ABCABC, in sq cm, is

  1. A.

    80

  2. B.

    68

  3. C.

    72

  4. D.

    78

Answer

C

Explanation

Let GG be the centroid of △ABC\triangle ABC. The centroid divides each median in the ratio 2:12:1. AG=23×12=8 cm,BG=23×9=6 cmAG = \frac{2}{3} \times 12 = 8\text{ cm}, \quad BG = \frac{2}{3} \times 9 = 6\text{ cm}

Since AD⊥BEAD \perp BE, △AGB\triangle AGB is a right-angled triangle with legs AG=8AG = 8 and BG=6BG = 6. Area(△AGB)=12×8×6=24 sq cm\text{Area}(\triangle AGB) = \frac{1}{2} \times 8 \times 6 = 24\text{ sq cm}

Since the centroid divides a triangle into three sub-triangles of equal area: Area(△ABC)=3×Area(△AGB)=3×24=72 sq cm\text{Area}(\triangle ABC) = 3 \times \text{Area}(\triangle AGB) = 3 \times 24 = 72\text{ sq cm}

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