Let's begin by understanding this geometric figure. We have four points labeled A, B, C, and D. The figure contains two right angles: one at point B and another at point A. We also have specific angle measurements: angle BAC is 60 degrees, and angle ADC is 45 degrees. The side BC has a length of 3 times the square root of 3.Let's focus on triangle ABC. This is a right triangle with the right angle at point B. We know that angle BAC is 60 degrees, which means angle ACB must be 30 degrees since the angles in a triangle sum to 180 degrees. The side BC has length 3 times the square root of 3. Using trigonometric ratios, we can determine the other sides of this triangle.Now let's calculate the length of side AB. In the right triangle ABC, we can use the tangent function. The tangent of 60 degrees equals BC divided by AB. Since tangent of 60 degrees is the square root of 3, and BC is 3 times the square root of 3, we can set up the equation: square root of 3 equals 3 times the square root of 3 divided by AB. Solving for AB, we get AB equals 3.Next, we'll find the length of the hypotenuse AC using the Pythagorean theorem. AC squared equals AB squared plus BC squared. Substituting our values: AC squared equals 3 squared plus 3 times the square root of 3 squared. This gives us 9 plus 27, which equals 36. Taking the square root of both sides, we find that AC equals 6.Now let's turn our attention to triangle ACD. This triangle has a right angle at point A. We're given that angle ADC is 45 degrees. Since the angles in a triangle must sum to 180 degrees, and we already have 90 degrees at A and 45 degrees at D, angle ACD must also be 45 degrees. This makes triangle ACD an isosceles right triangle, which means the two legs AD and AC are equal in length.In a 45-45-90 triangle, there's a special relationship between the sides. The two legs are always equal in length, and the hypotenuse is the square root of 2 times the length of each leg. Since we found that AC equals 6, and triangle ACD is a 45-45-90 triangle, we know that AD must also equal 6. The hypotenuse CD would be 6 times the square root of 2.Let's summarize all the key measurements we've found. In triangle ABC, we have AB equals 3, BC equals 3 times the square root of 3, and AC equals 6. In triangle ACD, we have AD equals 6 and CD equals 6 times the square root of 2. We also know all the angles: angle ABC is 90 degrees, angle DAB is 90 degrees, angle BAC is 60 degrees, and angle ADC is 45 degrees. This completely describes the geometric figure.This geometric figure beautifully demonstrates the properties of special right triangles. We have a 30-60-90 triangle ABC and a 45-45-90 triangle ACD sharing a common side AC. This configuration shows how different types of special triangles can be combined and how their properties relate to each other. Understanding these relationships is fundamental in geometry and trigonometry, and they appear frequently in various mathematical and real-world applications.