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 and the shortest side length is 2:1. Therefore, C = 2A.

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 . Therefore, .

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 . Therefore, .

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 . Therefore, .

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  is a diameter of the circle, arc  has a measure of 180 degrees. Therefore,  must be equal to . Since  is a right triangle, the sum of its interior angles to 180 degrees. Since the measure of  is twice the measure of , . Therefore, the measure of  can be calculated as follows:

Therefore,  is equal to .  must be a 30-60-90 triangle. Therefore, side length  must be half the length of side length , the hypotenuse of the triangle. Since  is a diameter of the circle, half of  represents the length of a radius of the circle. Therefore,  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  is a diameter of the circle, arc  has a measure of 180 degrees. Therefore, must be equal to . Since  is a right triangle, the sum of its interior angles equal 180 degrees. Since the measure of  is twice the measure of , . Therefore, the measure of  can be calculated as follows:

Therefore,  is equal to .  must be a 30-60-90 triangle.

Since  and  are perpendicular,  is a right angle. Since no triangle can have more than one right angle, and  is isosceles,  must be congruent to . Since  is congruent to  and  measures 90 degrees,  and  can be calculated as follows:

Therefore,    and  are both equal to 45 degrees.   is a 45-45-90 triangle. Since  is congruent to ,   is also a 45-45-90 triangle. The figure is drawn to scale, so  is a right angle. Since  has the same angle measure as , the two angles are alternate interior angles and diagonal  is a transversal relative to  and , which must be parallel.

Since  and  are perpendicular,  is a right angle. Since no triangle can have more than one right angle, and  is isosceles,  must be congruent to . Since angle CBD is congruent to  and  measures 90 degrees,  and  can be calculated as follows:

Therefore,  and  are both equal to 45 degrees.  is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse  is . Anyone of the four side lengths of quadrilateral  must, therefore, be equal to . To find the area of , multiply two side lengths: .