What is the area of the quadrilateral ABCD shown in the diagram below?

It should be C

Statement 1: Insufficient

We can measure of area of triangular region \(\triangle ABD\) but we have no idea of the position of \(C\), and thus no measure of the other \(\triangle BCD\).

Statement 1: Insufficient

We do not know any length

Statement 1 + 2: Sufficient

Let us consider two triangles \(\triangle ABD\) and \(\triangle BCD\). Let us calculate their areas separately.

\(\triangle ABD\):
From 1, we can calculate the area as well as the length \(BD\) using Pythagoras theorem knowing \(ABD\) is a right triangle. We also know the \(\angle ABD\) and \(\angle BDA\) are \(45\) given two sides \(AB\) and \(AD\) are equal.

\(\triangle BDC\):
From 2, we know \(\angle ADC\) is \(60\) degrees. We just calculated \(\angle BDA\) at \(45\). Thus,

\(\angle BDC = \angle ADC - \angle BDA = 60 - 45 = 15\).

\(\angle B\) is \(90\) (given) but its sub angle \(\angle ABD\) is \(45\). Thus we can safely say second sub angle of B i.e. \(\angle DBC\) is \(90 - 45 = 45\).

Knowing 2 angles and 1 side, we can accurately measure the remaining two sides and subsequently the area using Heron's formula.

\(Area = \sqrt{P(P-BD)(P-DC)(P-CB)}\) where P is half of perimeter of the \(\triangle BDC\).