The measure of ∠ACD in a circle is 90°. B is a point on the circle such that AB = BD. E is a point on AC such that ∠AEB = 90°. If AE = 5 cm and CE = 2 cm then find the length of CD.

To solve this problem, we will use the properties of circles and triangles, specifically the Pythagorean theorem and the property that the angle subtended by a diameter at the circumference is a right angle.

Step 1: Identify the given information. We know that ∠ACD = 90°, AB = BD, ∠AEB = 90°, AE = 5 cm, and CE = 2 cm.

Step 2: Use the Pythagorean theorem to find AC. Since ∠AEB = 90°, triangle AEB is a right triangle. Therefore, we can use the Pythagorean theorem (a² + b² = c²) to find AC. Here, AE is one leg of the triangle, CE is the other leg, and AC is the hypotenuse. So, AC = √(AE² + CE²) = √((5 cm)² + (2 cm)²) = √(25 cm² + 4 cm²) = √29 cm.

Step 3: Use the property of circles to find CD. Since ∠ACD = 90°, AD is a diameter of the circle. Therefore, CD, which is a radius of the circle, is half the length of AD. But since AB = BD, AD = 2AB = 2BD. Therefore, CD = AD/2 = AB = BD.

Step 4: Use the Pythagorean theorem to find BD. Since ∠ACD = 90°, triangle ACD is a right triangle. Therefore, we can use the Pythagorean theorem to find BD. Here, AC is one leg of the triangle, CD is the other leg, and AD is the hypotenuse. So, BD = √(AD² - AC²) = √((2BD)² - (√29 cm)²) = √(4BD² - 29 cm²).

Step 5: Solve the equation for BD. We have the equation √(4BD² - 29 cm²) = BD. Squaring both sides gives 4BD² - 29 cm² = BD². Simplifying gives 3*BD² = 29 cm², so BD = √(29 cm² / 3) = √(9.67 cm²) = 3.11 cm.

Step 6: Find CD. Since CD = BD, CD = 3.11 cm.

So, the length of CD is 3.11 cm.