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CAT 2022 Slot 3 — QA Question 17

Geometric CentersEasy

Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is

Answer & solution

Correct answer: 9

Answer: 9

Solution

Easy

The centroid OO splits △ABC\triangle ABC into 66 equal-area triangles of 1818 each. EE and DD are midpoints, so △AED∼△ABC\triangle AED\sim\triangle ABC with ratio 12\tfrac12, giving the area of quadrilateral EBCDEBCD. Subtract the four small triangles sitting inside EBCDEBCD to leave △EOD\triangle EOD.

A B C E D O
1

Six equal pieces at the centroid. The medians cut △ABC\triangle ABC into 66 triangles of equal area:

each=1086=18\text{each}=\frac{108}{6}=18
2

Area of trapezium EBCDEBCD. E,DE,D are midpoints of AB,ACAB,AC, so △AED∼△ABC\triangle AED\sim\triangle ABC with ratio 12\tfrac12:

[△AED]=(12)2×108=27[EBCD]=108−27=81\begin{aligned} &[\triangle AED]=\left(\tfrac12\right)^2\times 108=27\\ &[EBCD]=108-27=81 \end{aligned}
3

Carve out △EOD\triangle EOD. Quadrilateral EBCDEBCD contains △EOD\triangle EOD plus the four equal pieces △EOB,△BOF,△FOC,△COD\triangle EOB,\triangle BOF,\triangle FOC,\triangle COD:

[△EOD]=81−4×18=81−72=9[\triangle EOD]=81-4\times 18=81-72=9

Area of △EOD=9\triangle EOD=\mathbf{9} sq. cm.

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CAT 2022 Slot 3 QA Q17 — Geometric Centers solution | TheCATExam