Question

In △ABC, BD is an altitude.



What is the length, in units, of BD?

Answer 1

A formula that can be used is\[\frac{P_1}{a}=\frac{a}{P_2}\]Where P_1 is one of the segments that makes up the hypotenuse and P_2 is the other segment. Letting P_1=12 and P_2=15 we get\[\frac{12}{a}=\frac{a}{15}\]Cross-multiplying yields\[a^2=180\]Taking the square root of both sides gives us \[6\sqrt{5}\]We can ignore the negative result of the square root because we are dealing with distances, and distances cannot be negative.

Answer 2

Let BD = x , then
\[15^2 + x^2 = (AB)^2 and.. 12^2 + x^2 = (BC)^2\]
\[Also, In \triangle ABC , AB^2 + BC^2 = AC^2 =27^2 = 729 ..\]
\[Thus, By .. equating, (225+x^2) + (144+x^2) = 729\]
\[=> 2x^2 = 729 - 369 = 360\]
\[x^2 = 180 , => x=\sqrt{180} = 6\sqrt{5}\]