Question:
Euclid has a triangle in mind. Its longest side has length 20 and another of its sides has length 10. Its area is 80. What is the exact length of its third side?
Euclid has a triangle in mind. Its longest side has length 20 and another of its sides has length 10. Its area is 80. What is the exact length of its third side?
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Use area \(=\frac12 ab\sin\theta\) to find the angle between the sides 20 and 10, then use the cosine rule, keeping the root that makes 20 the longest side.
Updated On: Nov 4, 2026
- \(\sqrt{270}\)
- \(\sqrt{250}\)
- \(\sqrt{240}\)
- \(\sqrt{260}\)
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The Correct Option is D
Solution and Explanation
Step 1: Find the angle between the sides 20 and 10.
\[\tfrac12\cdot 20\cdot 10\cdot\sin\theta=80\Rightarrow \sin\theta=\tfrac45\Rightarrow \cos\theta=\pm\tfrac35\]Step 2: Apply the cosine rule for the third side \(x\).
\[x^2=20^2+10^2-2\cdot 20\cdot 10\cos\theta=500-400\cos\theta\]
The second case makes \(x\) larger than 20, so it is rejected. Therefore \(x^2=260\), \(x=\sqrt{260}\approx 16.1<20\).
Final Answer:
The correct option is (4): \(\sqrt{260}\).
\[\tfrac12\cdot 20\cdot 10\cdot\sin\theta=80\Rightarrow \sin\theta=\tfrac45\Rightarrow \cos\theta=\pm\tfrac35\]Step 2: Apply the cosine rule for the third side \(x\).
\[x^2=20^2+10^2-2\cdot 20\cdot 10\cos\theta=500-400\cos\theta\]
- \(\cos\theta=\frac35\): \(x^2=500-240=260\).
- \(\cos\theta=-\frac35\): \(x^2=500+240=740\), so \(x\approx 27.2>20\).
The second case makes \(x\) larger than 20, so it is rejected. Therefore \(x^2=260\), \(x=\sqrt{260}\approx 16.1<20\).
Final Answer:
The correct option is (4): \(\sqrt{260}\).
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