Short Answer 
The problem involves triangle ABC with a right angle at ACB and a 60° angle at ACD, leading to side CD calculated as 3√3 cm. The area of triangle ABC is calculated using the dimensions derived from its 45-45-90 triangle properties, confirming an area of 18√3 cm², which validates the calculations.
Step 1: Understanding the Triangle Configuration
In triangle ABC, we know that the angle ACB is a right angle (90°), and angle ACD is 60°. Consequently, angle BCD must be 30°. This configuration helps us determine side lengths using trigonometric functions. Specifically, we use the cosine function to calculate the length of side CD.
- Use the formula: cos(30°) = x/6
- Solving for x gives x = 6 * cos(30°)
- This results in x = 3‚àö3 cm for side CD.
Step 2: Calculating the Area of Triangle ABC
To find the area of triangle ABC, we can apply the formula for the area of a triangle, which is 1/2 * base * height. Since ABC is a special triangle with angles 45°-45°-90°, we can derive the lengths of the legs of the triangle based on side AC. Here, we take AC to be the base.
- Using the relationship of a 45-45-90 triangle, we find AC = 6‚àö3 cm.
- Height (BC) is also equal to the leg length, which is 6.
- This gives us: Area = 1/2 * 6‚àö3 * 6.
Step 3: Confirming the Area Calculation
Now, calculate the area using the derived dimensions. The area of triangle ABC can be confirmed using the dimensions calculated in step 2. This verification helps in ensuring that the area calculated initially aligns with the geometry of the triangle.
- Substituting the values, we find Area = 1/2 * 6√3 * 6 = 18√3 cm².
- This final area confirms the consistent properties of 45-45-90 triangles.
- Thus, we conclude that the area of triangle ABC = 18 cm² is valid under the given conditions.