Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the shortest side length and the longer non-hypotenuse side length is $\displaystyle 1:\sqrt{3}$. Therefore, $\displaystyle B=\sqrt{3}*A$.

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse length and the second-longest side length is $\displaystyle 2:\sqrt{3}$. Therefore, $\displaystyle 2B=\sqrt{3}*C$.

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the two shorter side lengths are equal. Therefore, A = B.

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between a short side length and the hypotenuse is $\displaystyle 1:\sqrt{2}$. Therefore, $\displaystyle C=\sqrt{2}*A$.

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between the two short side lengths is 1:1. Therefore, A = B. Triangles with two congruent side lengths are isosceles by definition.

For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because $\displaystyle AC$ is a diameter of the circle, arc $\displaystyle AC$ has a measure of 180 degrees. Therefore, $\displaystyle \angle ABC$ must be equal to $\displaystyle 180*\frac{1}2{}=90^{\circ}$. Since $\displaystyle \Delta ABC$ is a right triangle, the sum of its interior angles to 180 degrees. Since the measure of $\displaystyle \angle C$ is twice the measure of $\displaystyle \angle A$, $\displaystyle C=2A$. Therefore, the measure of $\displaystyle \angle A$ can be calculated as follows:

$\displaystyle A + B + C = 180$

$\displaystyle A + 90 + 2A = 180$

$\displaystyle 3A = 90$

$\displaystyle A = 30$

Therefore, $\displaystyle \angle C$ is equal to $\displaystyle 2*30=60^{\circ}$. $\displaystyle \Delta ABC$ must be a 30-60-90 triangle. Therefore, side length $\displaystyle \overline{BC}$ must be half the length of side length $\displaystyle \overline{AC}$, the hypotenuse of the triangle. Since $\displaystyle \overline{AC}$ is a diameter of the circle, half of $\displaystyle \overline{AC}$ represents the length of a radius of the circle. Therefore, $\displaystyle \overline{BC}$ is equal in length to a radius of the circle.

For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because $\displaystyle \overline{AC}$ is a diameter of the circle, arc $\displaystyle AC$ has a measure of 180 degrees. Therefore, $\displaystyle \angle ABC$must be equal to $\displaystyle 180*\frac{1}2{}=90^{\circ}$. Since $\displaystyle \Delta ABC$ is a right triangle, the sum of its interior angles equal 180 degrees. Since the measure of $\displaystyle \angle C$ is twice the measure of $\displaystyle \angle A$, $\displaystyle C=2A$. Therefore, the measure of $\displaystyle \angle A$ can be calculated as follows:

$\displaystyle A + B + C = 180$

$\displaystyle A + 90 + 2A = 180$

$\displaystyle 3A = 90$

$\displaystyle A = 30$

Therefore, $\displaystyle \angle C$ is equal to $\displaystyle 2*30=60^{\circ}$. $\displaystyle \Delta ABC$ must be a 30-60-90 triangle.

Since $\displaystyle \overline{BC}$ and $\displaystyle \overline{CD}$ are perpendicular, $\displaystyle \angle BCD$ is a right angle. Since no triangle can have more than one right angle, and $\displaystyle \Delta BCD$ is isosceles, $\displaystyle \angle CBD$ must be congruent to $\displaystyle \angle BDC$. Since $\displaystyle \angle CBD$ is congruent to $\displaystyle \angle BDC$ and $\displaystyle \angle BCD$ measures 90 degrees, $\displaystyle \angle CBD$ and $\displaystyle \angle BDC$ can be calculated as follows:

$\displaystyle BCD + CBD + BDC = 180$

$\displaystyle 90 + x + x = 180$

$\displaystyle 2x = 90$

$\displaystyle x = 45$ 

Therefore,  $\displaystyle \angle CBD$  and $\displaystyle \angle BDC$ are both equal to 45 degrees. $\displaystyle \Delta BCD$  is a 45-45-90 triangle. Since $\displaystyle \Delta ABD$ is congruent to $\displaystyle \Delta BCD$, $\displaystyle \Delta ABD$  is also a 45-45-90 triangle. The figure is drawn to scale, so $\displaystyle \angle DAB$ is a right angle. Since $\displaystyle \angle ADB$ has the same angle measure as $\displaystyle \angle DBC$, the two angles are alternate interior angles and diagonal $\displaystyle \overline{BD}$ is a transversal relative to $\displaystyle \overline{AD}$ and $\displaystyle \overline{BC}$, which must be parallel.

Since $\displaystyle BC$ and $\displaystyle CD$ are perpendicular, $\displaystyle \angle BCD$ is a right angle. Since no triangle can have more than one right angle, and $\displaystyle \Delta BCD$ is isosceles, $\displaystyle \angle CBD$ must be congruent to $\displaystyle \angle BDC$. Since angle CBD is congruent to $\displaystyle \angle BDC$ and $\displaystyle \angle BCD$ measures 90 degrees, $\displaystyle \angle CBD$ and $\displaystyle \angle BDC$ can be calculated as follows:

$\displaystyle BCD + CBD + BDC = 180$

$\displaystyle 90 + x + x = 180$

$\displaystyle 2x = 90$

$\displaystyle x = 45$

Therefore, $\displaystyle CBD$ and $\displaystyle BDC$ are both equal to 45 degrees. $\displaystyle \Delta BCD$ is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse $\displaystyle BD$ is $\displaystyle 1:\sqrt{2}$. Anyone of the four side lengths of quadrilateral $\displaystyle ABCD$ must, therefore, be equal to $\displaystyle \frac{BD}{\sqrt{2}}=\frac{1}{\sqrt{2}}$. To find the area of $\displaystyle ABCD$, multiply two side lengths: $\displaystyle \frac{1}{\sqrt{2}}*\frac{1}{\sqrt{2}}=\frac{1}{2}$.