The third side

Geometry Level 3

A triangle with side lengths 9 9 and 44 44 has an area of 198 198 . If the length of the third side is in the form a \sqrt{a} , submit your answer as a a .


The answer is 2017.

Sathvik Acharya by Sathvik Acharya , 17, India

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1 solution

Sathvik Acharya
May 18, 2017

Observe that 9 × 44 2 \dfrac{9\times 44}{2} = 198 =198 , which implies that the triangle is right angled. By the Pythagorean theorem (since it is a right angled triangle and our third side is the hypotenuse) the third side is 9 2 + 4 4 2 = 2017 \sqrt{9^2+44^2}=\sqrt{2017} . Therefore, a = 2017 a=2017

2 Helpful 0 Interesting 0 Brilliant 0 Confused

the observation that it is right angled is just the sin rule in disguise.

Ayush G Rai - 4 years ago

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