CAT 2022 Geometry & Mensuration Question (Medium)

Question

The length of each side of an equilateral triangle is . Let be a point on such that the area of triangle is half the area of triangle . Then the length of , in , is

Accepted Answer

Step 1 Let triangle be equilateral with side length 3 cm. Let , , and using coordinate geometry. Let point lie on . Parameterize as where is between 0 and 1. Step 2 Area of triangle Area of triangle Given Area of triangle Area of triangle So 1 2 x_D y_C - x_C y_D 3 4 y_D Substitute , , , Now compute

CAT 2022

Quantitative

Geometry & Mensuration

Medium

The length of each side of an equilateral triangle $\mathrm{ABC}$ is $3 \mathrm{~cm}$ . Let $\mathrm{D}$ be a point on $\mathrm{BC}$ such that the area of triangle $\mathrm{ADC}$ is half the area of triangle $\mathrm{ABD}$ . Then the length of $\mathrm{AD}$ , in $\mathrm{cm}$ , is

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