Geom Question bank query

[jigar24]

In the Geom question bank, for DS question

If angle BAD is a right angle, what is the length of side BD?
(1) AC is perpendicular to BD

(2) BC = CD

Solution says:
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).

How can we say it’s an isosceles triangle? Is this a property? Please explain

Jigar

[Pankaj.Dahiya]

Lets call AC=h and BC=CD=x
Using Pythagoras, AB=AD=x^2+h^2

[j.vishal]

As AC bisects BD, it would bisect angle BAD as well. So angle BAC= angle CAD = 45. As the triangles ABC and CAD are also rightangle triangles, so angle CBA and angle CDA would be 45 each.

The side of the triangle facing the equal angles are also same so BC=CD=AB=AD. Hence Triangle BAD is an isosceles triangle.

Thanks

[Ben Ku]

Thanks j.vishal. That was a good explanation.

Another approach is to say that both right triangles ACB and ACD have two congruent legs, (AC = AC and CB = CD), then their hypotenuses are also congruent (AB = AD). BAD is isosceles.

Hope that helps!