Question

In the right triangle ABC with ∠C = 90°, ∠B = 75°, and AB = 12 cm, find the area of triangle ABC.

a) 36√3 cm²
b) 54√3 cm²
c) 72√3 cm²
d) 108√3 cm²

Answer 1

The area of the right triangle ABC can be determined by calculating the lengths of sides BC and AC using the hypotenuse AB and the known angle B, and then applying the formula for the area of a triangle (1/2 × base × height).

Step-by-step explanation:

In the given right triangle ABC with ∠C = 90°, ∠B = 75°, and AB = 12 cm, we can find the lengths of the other two sides using trigonometric ratios, after which we can calculate the area. First, since AB is the hypotenuse, we'll find the length of BC (adjacent to ∠B) using the cosine:

BC = AB × cos(∠B) = 12 cm × cos(75°)

Then, using the sine function, we determine AC (opposite to ∠B):

AC = AB × sin(∠B) = 12 cm × sin(75°)

After calculating these lengths, we know that the area of a triangle is given by:

Area = ½ × base × height

In a right triangle, the base and height correspond to the two legs, so we have:

Area = ½ × BC × AC

By substituting the calculated lengths, we can find the exact area of the triangle ABC.