Video Transcript
𝐴𝐵𝐶𝐷 is a quadrilateral where 𝐴𝐵 equals 31 centimeters, 𝐵𝐶 equals 40 centimeters, 𝐶𝐷 equals 43 centimeters, 𝐷𝐴 equals 32 centimeters, and the measure of angle 𝐴 equals 64 degrees. Find the measure of angle 𝐶𝐵𝐷, giving the answer to the nearest degree.
Let’s begin by sketching this quadrilateral. We’ll start with angle 𝐴, which we know to be 64 degrees, and the two sides which enclose this angle: 𝐴𝐵, which is 31 centimeters, and 𝐷𝐴 or 𝐴𝐷, which is 32 centimeters. We’ll then add in vertex 𝐶 and the two sides that enclose this angle: 𝐵𝐶, which is 40 centimeters, and 𝐶𝐷, which is 43 centimeters. So we have a sketch of our quadrilateral. We’re asked to find the measure of angle 𝐶𝐵𝐷. That’s the angle formed when we travel from 𝐶 to 𝐵 to 𝐷. If we add in the straight line connecting vertices 𝐵 and 𝐷, then we’re looking for the measure of this angle here. Let’s think about how we could calculate the measure of this angle.
This angle belongs in triangle 𝐵𝐶𝐷 in which the only information we know are two of the triangle’s side lengths. This isn’t enough for us to be able to calculate the measure of any angle, so let’s consider what other information we have. This line 𝐵𝐷 divides the quadrilateral into two triangles: triangle 𝐵𝐶𝐷, which we’ve already discussed, and triangle 𝐴𝐵𝐷. The side 𝐵𝐷 is common to both triangles, so it may be possible to calculate the length of 𝐵𝐷 using information in triangle 𝐴𝐵𝐷. That’s the green triangle. In this triangle, we note that we’ve been given the length of two of the sides and the measure of their included angle. This means that we can calculate the length of the third side of this triangle by applying the law of cosines.
This states that if we know the length of two sides in any triangle, which we refer to as 𝑏 and 𝑐, and the measure of their included angle, which is uppercase 𝐴, then we can calculate the length of the third side, that’s lowercase 𝑎 which is opposite angle 𝐴, using the formula 𝑎 squared equals 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴. For our triangle, 𝐵𝐷 is the unknown side opposite the known angle of 64 degrees. So 𝐵𝐷 is side 𝐴 in this formula. The sides which enclose this angle are 𝐴𝐵 and 𝐴𝐷, which are 31 and 32 centimeters, respectively. So these are the lengths of 𝑏 and 𝑐. And then the enclosed angle is 64 degrees. So we have 𝐵𝐷 squared equals 31 squared plus 32 squared minus two times 31 times 32 times cos of 64 degrees. And this is an equation we can solve to calculate the length of 𝐵𝐷.
Evaluating parts of this expression gives 𝐵𝐷 squared equals 961 plus 1024 minus 1984 cos of 64 degrees. Evaluating further, we have that 𝐵𝐷 squared is equal to 1115.271 continuing. 𝐵𝐷 is then the square root of this value, which is 33.395 continuing.
So, using the information we were given in triangle 𝐴𝐵𝐷, we’ve been able to calculate the length of the side that is shared between the two triangles. This means that in triangle 𝐵𝐶𝐷, we now know the length of all three of this triangle sides, and we want to calculate the measure of one of its angles. We can do this using another application of the law of cosines. The law of cosines for calculating an angle, which is just a direct rearrangement of the law of cosines we’ve already seen, states that the cos of an angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all over two 𝑏𝑐.
So we can calculate the measure of any angle in a triangle if we know the length of all three sides. Here, side 𝑎 is the side opposite the angle we want to calculate. So in our triangle, that’s the side of length 43 centimeters. And the sides which enclose the angle are 𝐵𝐶, which is 40 centimeters, and 𝐵𝐷, which is 33.395 centimeters. So, we can substitute 40 for side 𝑏 throughout this formula. In the numerator, we then have 𝑐 squared, so that is side 𝐵𝐷 squared, and we can substitute the value before we square rooted. That’s 1115.271. In the denominator, where we have 𝑐, we substitute 33.395. That’s the value for 𝐵𝐷 once we had square rooted. And finally in the numerator, we’re subtracting 𝑎 squared. So we subtract 43 squared.
Evaluating this using values which are as accurate as possible for 𝐵𝐷 squared and 𝐵𝐷 gives 0.324 continuing. To find the measure of angle 𝐶𝐵𝐷, we then apply the inverse cosine function and evaluate. This gives 71.0 continuing, which to the nearest degree is 71 degrees. So, by applying the law of cosines twice, once to calculate the length of an unknown side and once to calculate the measure of an unknown angle, we found that the measure of angle 𝐶𝐵𝐷 to the nearest degree is 71 degrees.
