𝐴𝐵𝐶𝐷 is a quadrilateral where 𝐴𝐵 is equal to 10 centimeters, the measure of angle 𝐴𝐷𝐵 is 74 degrees, the measure of angle 𝐴𝐵𝐷 is 32 degrees, 𝐵𝐶 is equal to 16 centimeters, and 𝐷𝐶 is equal to 18 centimeters. Find the measure of angle 𝐶𝐵𝐷, giving the answer to the nearest second.
We will begin by sketching the quadrilateral and labeling the sides and angles we’re given. We are told that 𝐴𝐵 is equal to 10 centimeters, 𝐵𝐶 is equal to 16 centimeters, and 𝐷𝐶 is equal to 18 centimeters. The measure of angle 𝐴𝐷𝐵 is 74 degrees, and the measure of angle 𝐴𝐵𝐷 is 32 degrees. We are asked to calculate the measure of angle 𝐶𝐵𝐷, which we will call 𝜃.
If we begin by considering triangle 𝐴𝐵𝐷, we know that angles in a triangle sum to 180 degrees. This means that to calculate the measure of angle 𝐵𝐴𝐷, we can subtract 32 degrees and 74 degrees from 180 degrees. This gives us an answer of 74 degrees. As two of the angles in the triangle are equal, we have a an isosceles triangle, which means that the lengths of two sides will also be equal. The lengths of the sides 𝐴𝐵 and 𝐷𝐵 are both equal to 10 centimeters.
Let’s now consider the triangle 𝐵𝐶𝐷. We know the lengths of all three sides, and we are trying to calculate one of the angles in the triangle. We can do this using the law of cosines or cosine rule, which states that the cos of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all divided by two 𝑏𝑐. Note that the side length we are subtracting must be opposite our angle, in this case, 18 centimeters.
Substituting in our values, we have the cos of angle 𝜃 is equal to 16 squared plus 10 squared minus 18 squared all divided by two multiplied by 16 multiplied by 10. The right-hand side simplifies to one over 10. The cos of angle 𝜃 is equal to one-tenth. We can then take the inverse cos of both sides. 𝜃 is equal to the inverse cos of one-tenth. 𝜃 is therefore equal to 84.2608 and so on degrees. We are asked to give our answer to the nearest second. So 𝜃 is equal to 84 degrees, 15 minutes, and 39 seconds. This is the measure of angle 𝐶𝐵𝐷 to the nearest second.