Final answer:
To find the perimeter of △ABC, calculate the lengths of all three sides using the given angles and the fact that △ABC is a right triangle. Then add the lengths of all three sides together. To find the area of △ABC, use the formula A = 1/2 * base * height. Choose a base and height from the triangle and substitute the known values into the formula to calculate the area.
Step-by-step explanation:
To find the perimeter of △ABC, we need to calculate the lengths of all three sides and add them together. Given that AB = 9, we need to find the lengths of BC and CA. Since the sum of the angles in a triangle is always 180 degrees, we can use the angles given and the fact that △ABC is a right triangle to find the missing angles. Since m∠A = 60° and m∠C = 45°, m∠B = 180° - 60° - 45° = 75°. Since △ABC is a right triangle, we can use the sine and cosine functions to find the lengths of the sides. Using sine, we have sin∠A = BC / AB. Substituting the known values, we have sin(60°) = BC / 9. Solving for BC, we get BC = 9 * sin(60°). Similarly, using cosine, we have cos∠A = CA / AB. Substituting the known values, we have cos(60°) = CA / 9. Solving for CA, we get CA = 9 * cos(60°). With the lengths of all three sides, we can calculate the perimeter by adding them together.
To find the area of △ABC, we can use the formula A = 1/2 * base * height, where the base is one of the sides of the triangle and the height is the perpendicular distance from the base to the opposite vertex. Since △ABC is a right triangle, the base can be AB or BC, and the height can be CA or BC. We can choose any combination of base and height. Let's use AB as the base and CA as the height. Substituting the known values, we have A = 1/2 * 9 * (9 * cos(60°)). Simplifying, we have A = 1/2 * 9 * 9 * 0.5, which equals 40.5 square units.