The sides x and y of a scalene triangle satisfy x+(2 Δ)/(x)=y+(2 Δ)/(y) , where Δ is the area of the triangle. If x=60, y=63 , what is the length of the largest side of the triangle?

Correct answer: 87 aligned & 60+(2 Δ)/(60)=63+(2 Δ)/(63) & Δ=1890 aligned Let third side be c and angle C aligned & (1)/(2) · 60 · 63 · C=1890 & C=1 C=(π)/(2) aligned A B C is right triangle Longest side is hypotenuse =60^2+63^2 aligned & z^2=7569 & z=87 largest side. aligned

Olympiad workbook

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

Olympiad workbook · IOQM · Properties of Triangles

The sides x and y of a scalene triangle satisfy x+(2 Δ)/(x)=y+(2...

The sides

x

and

y

of a scalene triangle satisfy

x+\frac{2 \Delta}{x}=y+\frac{2 \Delta}{y}

, where

\Delta

is the area of the triangle. If

x=60, y=63

, what is the length of the largest side of the triangle?

Correct answer

Step-by-step solution

\begin{aligned} & 60+\frac{2 \Delta}{60}=63+\frac{2 \Delta}{63} \\ & \Rightarrow \Delta=1890 \end{aligned}

Let third side be

c

and angle

C

\begin{aligned} & \Rightarrow \frac{1}{2} \cdot 60 \cdot 63 \cdot \sin C=1890 \\ & \sin C=1 \quad \Rightarrow C=\frac{\pi}{2} \end{aligned}

\Rightarrow \triangle A B C

is right triangle Longest side is hypotenuse

=60^2+63^2

\begin{aligned} & \Rightarrow z^2=7569 \\ & \Rightarrow z=87 \text { largest side. } \end{aligned}

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