Video Transcript
An isosceles triangle has two sides of length 48 centimeters and base angles of 73 degrees. Find the area of the triangle, giving the answer to three decimal places.
Let’s begin by sketching this triangle. We’re told that it is an isosceles triangle and it has two sides of length 48 centimeters. So these are the two sides of equal length. The base angles, which are the angles between each of these sides and the third side of the triangle, are each 73 degrees. The triangle therefore looks like this. We’re asked to find the area of this triangle.
Now, usually, we might find the area of a triangle using the formula base multiplied by perpendicular height over two. But in this problem, we don’t know either the base or the perpendicular height of this triangle. We could calculate each of these values using trigonometry. But there is in fact a more straightforward way to find the area of this triangle.
We can recall the trigonometric formula for the area of a triangle. If we have a triangle 𝐴𝐵𝐶, where the uppercase letters 𝐴, 𝐵, and 𝐶 represent the vertices of the triangle and the lowercase letters 𝑎, 𝑏, and 𝑐 represent the side lengths opposite each of these angles, the area of this triangle is given by the formula a half 𝑎𝑏 sin 𝐶. The letters 𝑎 and 𝑏 represent the lengths of any two sides in the triangle. And the uppercase 𝐶 represents the measure of their included angle. That’s the angle between the two side lengths 𝑎 and 𝑏.
Returning to the isosceles triangle in this question, we are given the lengths of two sides, the two equal sides of the isosceles triangle, which are each of length 48 centimeters. We don’t yet know the measure of their included angle, but we can work it out. We know that the angles in any triangle sum to 180 degrees. So we can calculate the measure of the third angle by subtracting the measures of the other two, the two base angles, from 180 degrees. We have 180 degrees minus 73 degrees minus 73 degrees, which is 34 degrees.
We now know the lengths of two sides in this triangle and the measure of their included angle. So we can substitute these values into the trigonometric formula for the area of a triangle. We have that the area is equal to a half multiplied by 48 multiplied by 48 multiplied by sin of 34 degrees. That gives 1152 multiplied by sin of 34 degrees. And we can then evaluate this on a calculator, ensuring that it is in degree mode. That gives 644.19022 continuing.
The question specifies that we should give our answer to three decimal places. As the digit in the fourth decimal place is a two, we round down to 644.190. The units for this area are square centimeters. So by applying the trigonometric formula for the area of a triangle, we found that the area of this isosceles triangle to three decimal places is 644.190 square centimeters.
