Let A B C be an equilateral triangle with side length 10. A square P Q R S is inscribed in it, with P on A B , Q, R on B C and S on A C . If the area of the square P Q R S is m+n k where m, n are integers and k is a

Correct answer: 10 aligned & a+2 x=10 & also 60^ =(a)/(x) & x= a 3 & a= 10 3 2+ 3 & Area of square =a^2=300(7-4 3 ) & m=2100, n=-1200, k=3 & (m+n)/(k^2) = (900)/(9) =10 aligned

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

Let A B C be an equilateral triangle with side length 10. A square...

Let

A B C

be an equilateral triangle with side length 10.

A

square

P Q R S

is inscribed in it, with

P

A B

Q, R

B C

and

S

A C

. If the area of the square

P Q R S

m+n \sqrt{k}

where

m, n

are integers and

k

is a prime number then determine the value of

\sqrt{\frac{m+n}{k^2}}

Let A B C be an equilateral triangle with side length 10. A square P Q R S is inscribed in it, with P on A B , Q, R on

Correct answer

Step-by-step solution

\begin{aligned} & \therefore a+2 x=10 \\ & \text { also } \tan 60^{\circ}=\frac{a}{x} \\ & \Rightarrow x=\frac{a}{\sqrt{3}} \\ & \therefore a=\frac{10 \sqrt{3}}{2+\sqrt{3}} \\ & \text { Area of square }=a^2=300(7-4 \sqrt{3}) \\ & \therefore m=2100, n=-1200, k=3 \\ & \sqrt{\frac{m+n}{k^2}}=\sqrt{\frac{900}{9}}=10\end{aligned}

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