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If altitude CD is sqrt3 centimeters, what is  BD?

 

 Nov 4, 2020
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We have a 30 60 90 triangle.

Let $BC = x.$ $AC = x \sqrt{3},$ and $AB = 2x.$

We also have $AC * CB = \sqrt{3} * AB.$

Let $DB = y.$ Then $$y^2+3=x^2,$$ and $$3+(2x-y)^2 = (x \sqrt {3})^2.$$ (by the pythagorean theorem.)

 

Expanding, we have $$y^2+3=x^2,$$ and $$4 x^2 - 4 x y + y^2 + 3 = 3 x^2.$$

 

If we solve this equation, we get $x=2$ and $y=1,$ for integer solutions. Thus $BD = y = 1.$

 Nov 4, 2020

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