Geometry

[gmatusa2010]

Picture: https://tinyurl.com/33lglbj


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

(1) AC is perpendicular to BD

(2) BC = CD

(1) AND (2) SUFFICIENT: 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.

Can someone explain the bolded part?

[kvcpk]

This is more of a Geometry rule. So you will need to remember this.
in an ISOCELES triangle, the Perpendicular from the vertex bisects the non-equal side.
Opposite is also true.

Hence, Combinign both statements, we know that AC is perpendicular and that it bisects the non-equal side. Hence it should be an Isoceles triangle.

Its good to remember this rule. Just in case you want to know the proof:

Let triangle PQR, PQ = PR and QR is non equal side.
Now, Draw a perpendicular from P to line QR.
Let us name the point on QR as S.
now, Triangles, PQS and PRS are Congruent triangles.
Why?
becasue angle(PSQ)=angle(PSR), angle(PQS)=angle(PRS) and PQ = PR
by SAS property, both triangles are congruent.

Hence, QS = RS


Hope this helps!!