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CAT 2019 Slot 2 — QA Question 4

Geometric CentersEasy

In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, is

Answer & solution

Correct answer: 72

  • A

    80

  • B

    68

  • 72

  • D

    78

Solution

Easy

The centroid GG divides each median in the ratio 2:12:1, so AG=8AG=8, BG=6BG=6. Because the medians are perpendicular, triangle ABGABG is right-angled at GG. The three medians cut the triangle into six small triangles of equal area, and △ABC=3 △ABG\triangle ABC = 3\,\triangle ABG.

A B C D E G
1

Split each median in ratio 2:12:1 at the centroid. With AD=12AD=12 and BE=9BE=9:

AG=23×12=8,GD=4BG=23×9=6,GE=3\begin{aligned} &AG=\tfrac{2}{3}\times 12=8,\quad GD=4\\ &BG=\tfrac{2}{3}\times 9=6,\quad GE=3 \end{aligned}
2

Area of right triangle ABGABG. The medians are perpendicular at GG, so △ABG\triangle ABG is right-angled there with legs AG=8AG=8 and BG=6BG=6 from step 1.

[△ABG]=12×8×6=24\begin{aligned} &[\triangle ABG]=\tfrac{1}{2}\times 8\times 6=24 \end{aligned}
3

Scale up to the whole triangle. The medians from AA and BB make [△ABC]=3 [△ABG][\triangle ABC]=3\,[\triangle ABG].

[△ABC]=3×24=72\begin{aligned} &[\triangle ABC]=3\times 24=72 \end{aligned}
Area of △ABC=72 sq cm\text{Area of }\triangle ABC=72\ \text{sq cm}

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CAT 2019 Slot 2 QA Q4: In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respect — Solution | TheCATExam