Isosceles triangle length

Geometry Level 1

Above is a isosceles triangle. Which have an area of 3000 cm 2 \text{cm}^2 . Find c c


The answer is 85.

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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2 solutions

Munem Shahriar
Feb 3, 2018

The area of the isosceles triangle is b 4 4 c 2 − b 2 \dfrac b4 \sqrt{4c^2 - b^2} , where b = 80 cm b = 80 \text{cm} .

b 4 4 c 2 − b 2 = 3000 80 4 4 c 2 − 8 0 2 = 3000 20 4 c 2 − 8 0 2 = 3000 4 c 2 − 8 0 2 = 150 ( 4 c 2 − 6400 ) 2 = ( 150 ) 2 [ Square on both sides ] 4 c 2 − 6400 = 22500 4 c 2 = 28900 c 2 = 7225 ⟹ c = 85 \begin{aligned} \dfrac b4 \sqrt{4c^2 - b^2} & = 3000 \\ \dfrac{80}{4} \sqrt{4c^2 - 80^2} & = 3000 \\ 20 \sqrt{4c^2 - 80^2} & = 3000 \\ \sqrt{4c^2 - 80^2} & = 150 \\ (\sqrt{4c^2 - 6400})^2 & = (150)^2 ~~~~~~~~~~~~ [\text{Square on both sides}] \\ 4c^2 - 6400 & = 22500 \\ 4c^2 & = 28900 \\ c^2 & = 7225 \\ \implies c&=\boxed{85} \\ \end{aligned}

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You can get the answer in 2 steps. Draw a median from A to BC. Since, it is an isosceles triangle, the median is also the perpendicular bisector. Using half base into height formula, find the height. After that, use pythagoras theorem to find c.

A Former Brilliant Member - 3 years, 4 months ago
Mahdi Raza
May 4, 2020

1 2 b h = 3000 ⟹ h = 75 \frac{1}{2}bh = 3000 \implies h = 75 . By Pythagorean theorem: c 2 = 7 5 2 + 4 0 2 c^2 = 75^2 + 40^2 ⟹ c = 85 \implies \boxed{c = 85}

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