Area of triangle ratio

CAT 2021 Slot 2 · QA · Medium · Geometry

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Let DD and EE be points on sides ABAB and ACAC, respectively, of a triangle ABCABC, such that AD:BD=2:1AD : BD = 2 : 1 and AE:CE=2:3AE : CE = 2 : 3. If the area of the triangle ADEADE is 8 sq cm, then the area of the triangle ABCABC, in sq cm, is

Answer

30

Explanation

In △ABC\triangle ABC, AD:AB=2:3AD : AB = 2 : 3 and AE:AC=2:5AE : AC = 2 : 5.

The area of △ADE\triangle ADE is given by: Area(△ADE)=12⋅AD⋅AE⋅sin⁡A\text{Area}(\triangle ADE) = \frac{1}{2} \cdot AD \cdot AE \cdot \sin A Area(△ABC)=12⋅AB⋅AC⋅sin⁡A\text{Area}(\triangle ABC) = \frac{1}{2} \cdot AB \cdot AC \cdot \sin A

Ratio of areas: Area(△ADE)Area(△ABC)=ADAB×AEAC=23×25=415\frac{\text{Area}(\triangle ADE)}{\text{Area}(\triangle ABC)} = \frac{AD}{AB} \times \frac{AE}{AC} = \frac{2}{3} \times \frac{2}{5} = \frac{4}{15}

Given Area(△ADE)=8\text{Area}(\triangle ADE) = 8: 8Area(△ABC)=415  ⟹  Area(△ABC)=30 sq cm.\frac{8}{\text{Area}(\triangle ABC)} = \frac{4}{15} \implies \text{Area}(\triangle ABC) = 30\text{ sq cm.}

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