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CAT 2020 Slot 2 — QA Question 23

Basics of TrianglesEasy

From an interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the lengths of the three perpendiculars is s. Then the area of the triangle is

Answer & solution

  • S2/√3

  • B

    √3S2/2 

  • C

    S2/2√3

  • D

    2S2/√3

Solution

Easy

Join the interior point to the three vertices. This splits the equilateral triangle into three smaller triangles whose heights are exactly the three perpendiculars. Their areas must add up to the whole, which links the side aa to the sum ss (Viviani's theorem).

P h1 a
1

Split the area. Let the side be aa and the three perpendiculars be h1,h2,h3h_1,h_2,h_3, so h1+h2+h3=sh_1+h_2+h_3=s. The three small triangles share the same base aa and tile the whole triangle:

12ah1+12ah2+12ah3=34a2(parts = whole)⇒ 12a (h1+h2+h3)=34a2⇒ 12a s=34a2(sum of perpendiculars =s)\begin{aligned} &\tfrac12 a h_1+\tfrac12 a h_2+\tfrac12 a h_3=\frac{\sqrt3}{4}a^2 \quad\text{(parts = whole)}\\ &\Rightarrow\ \tfrac12 a\,(h_1+h_2+h_3)=\frac{\sqrt3}{4}a^2\\ &\Rightarrow\ \tfrac12 a\,s=\frac{\sqrt3}{4}a^2 \quad\text{(sum of perpendiculars }=s) \end{aligned}
2

Solve for the side. Cancel one factor of aa from step 1:

12s=34a⇒ a=2s3\begin{aligned} &\tfrac12 s=\frac{\sqrt3}{4}a\\ &\Rightarrow\ a=\frac{2s}{\sqrt3} \end{aligned}
3

Compute the area. Substitute a=2s3a=\dfrac{2s}{\sqrt3} into the area formula:

Area=34a2=34⋅4s23(from step 2)⇒ Area=s23\begin{aligned} &\text{Area}=\frac{\sqrt3}{4}a^2=\frac{\sqrt3}{4}\cdot\frac{4s^2}{3} \quad\text{(from step 2)}\\ &\Rightarrow\ \text{Area}=\frac{s^2}{\sqrt3} \end{aligned}
Area=s23\text{Area}=\dfrac{s^2}{\sqrt3}

Viviani's theorem: for any interior point of an equilateral triangle, h1+h2+h3h_1+h_2+h_3 equals the triangle's altitude H=32aH=\frac{\sqrt3}{2}a. So s=32a⇒a=2s3s=\frac{\sqrt3}{2}a\Rightarrow a=\frac{2s}{\sqrt3}, giving area =s23=\frac{s^2}{\sqrt3} directly.

CAT 2020 Slot 2 QA Q23: From an interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the — Solution | TheCATExam