Problem of the Day

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.  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, $AD=x=4$.