MGMAT Question Bank - Geometry #19

Image: A right triangle BAD (with right angle A) has a segment AC drawn from vertex A to side BD. AD is labeled as 5.

If angle BAD is a right angle, what is the length of side BD?

(1) AC is perpendicular to BD

(2) BC = CD

The answer is C. The explanation tells us: Using statements 1 and 2, we know that AC is the perpendicular bisector of BD. This means that triangle BAD is an isosceles triangle so side AB must have a length of 5 (the same length as side AD). We also know that angle BAD is a right angle, so side BD is the hypotenuse of right isosceles triangle BAD. If each leg of the triangle is 5, the hypotenuse (using the Pythagorean theorem) must be 5.

I didn't understand why AC being the perpendicular bisector of BD makes the triangle isosceles. Is this a rule that we should know? Thanks!

Yep, something you need to know. If you drop a line from the right angle of a triangle to the opposite side, and if that line both creates a right angle and bisects the other side, then it always means that the two sides on either side of the original right angle are the same length. And if two sides are the same length, then the triangle is isosceles.

The proof (if you care to know!) is that if you create the above situation, you know now that two of the sides are the same and the angle between them is the same. SAS (or side angle side) is one of the proofs we can use to demonstrate congruent triangles. And if the two "inner" triangles are congruent, then those two outer sides by the right angle of the big triangle are also the same length... hence isosceles.