# Problem of the Day

In the above triangle, if , what is the length of ?

(1)

(2)

D. Each statement alone is sufficient.

See the Solution

### Solution

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Statement 1: SUFFICIENT. If we know that $BD=2\sqrt{5}$ then we can use the Pythagorean theorem to determine the length of $\overline{AD}$.

Statement 2: SUFFICIENT. When a perpendicular line segment is drawn from the right angle of a right triangle to the hypotenuse, it creates two new right triangles. This effectively creates three pairs of similar right triangles: the new right triangles are similar to each other, and each new right triangle is similar to the original right triangle. Â Using this fact, we can set up the following proportion, where $AD=x$:

$\displaystyle \frac{6}{x}=\frac{5+x}{6}$

$x(x+5)=36$

$x^2+5x-36=0$

$(x-4)(x+9)=0$

$x=4, x=-9$

$x$ represents the length of a line segment, so it cannot be negative. Hence, $AD=x=4$.