chiragatara wrote:

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

(1) AC is perpendicular to BD

(2) BC = CD

Official answer explanation:

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 underroot 2.

Can someone kindly explainthe underlined portion?

The goal here is to find the length of side BD. Since it's DS, the problem is not drawn to scale.

Statement 1) AC is perpendicular to BD.

We don't know if BC = BD, so this statement is insufficient. There are many possible triangles here that would give a different length of BD.

Statement 2) BC = CD

We don't know if AC is perpendicular to BD, so knowing that the two sides are equal doesn't help us.

Statements 1+2) We know that AC is the perpendicular bisector of BD, so that means that the line AC forms two isosceles triangles (ACD and ACB). Since the sides of the height is equal to their bases and two each other, then that means the two triangles are congruent. Since they are congruent, BAD is an isosceles triangle and its base, the hypotenuse of the triangle in this case, is 5 \sqrt{2}