Is it that far?

Geometry Level 2

The tower for a radio station stands 200 200 ft tall. A guy wire 250 250 ft long supports the antenna on one side and it is fastened at point A A on the ground. Another guy wire 300 300 ft long supports the antenna on the opposite side and it is fastened at point B B on the ground. Assuming that the ground is horizontal, how far is A A from B B in feet?

150 + 100 5 150+100\sqrt{5} 2500 + 500000 \sqrt{2500}+\sqrt{500000} 370 370 150 + 5000 150+\sqrt{5000}

A Former Brilliant Member by A Former Brilliant Member

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1 solution

By pythagorean theorem , we have

( A D ) 2 = 25 0 2 − 20 0 2 = 22500 (AD)^2=250^2-200^2=22500

A D = 22500 = 150 AD=\sqrt{22500}=150

By pythagorean theorem again, we have

( D G ) 2 = 30 0 2 − 20 0 2 = 50000 (DG)^2=300^2-200^2=50000

D G = 50000 = 100 5 DG=\sqrt{50000}=100\sqrt{5}

Finally,

A B = A D + D B = AB = AD+DB= 150 + 100 5 \large\color{#3D99F6}\boxed{150+100\sqrt{5}} answer \color{#69047E}\boxed{\text{answer}}

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