Video Transcript
𝐴𝐵𝐶𝐷 is a rhombus whose diagonals intersect at the point 𝑀, where 𝐴𝐵 equals 11 centimeters and 𝐴𝑀 equals 10 centimeters. Find the value of angle 𝐵𝐴𝐷, giving the answer to the nearest second.
We begin by recalling that the diagonals of a rhombus bisect each other and intersect at right angles as shown. This means that the rhombus has four congruent triangles, of which 𝑀𝐴𝐵 is one. We are told that 𝐴𝐵 is of length 11 centimeters and 𝐴𝑀 is of length 10 centimeters. We are asked to find the value of the angle 𝐵𝐴𝐷. Since the triangles are congruent, the measure of this angle will be equal to two multiplied by the measure of angle 𝐵𝐴𝑀.
We will begin by trying to calculate the measure of this angle labeled 𝜃. As we have a right triangle, we can use our knowledge of the trigonometric ratios. These state that sin 𝜃 is equal to the opposite over the hypotenuse. cos 𝜃 is equal to the adjacent over the hypotenuse. And tan 𝜃 is equal to the opposite over the adjacent. We can remember these ratios using the acronym SOH CAH TOA. The hypotenuse is the longest side of a right triangle. The opposite side is the side that is opposite the angle we are working with. And the adjacent is the side that is next to this angle and the right angle.
In this question, the length 𝐴𝑀 is the adjacent and is equal to 10 centimeters. The hypotenuse 𝐴𝐵 is of length 11 centimeters. We will therefore use the cosine ratio to help us calculate the value of angle 𝜃. Substituting in these values, we have cos 𝜃 is equal to 10 over 11. We can then take the inverse cosine of both sides. 𝜃 is equal to the inverse cos of 10 over 11. Ensuring our calculator is in degree mode, typing the right-hand side gives us 24.6199 and so on degrees. We can then use the degrees, minutes, seconds button to give our answer to the nearest second. Noting that there are 60 minutes in a degree and 60 seconds in a minute, this gives us 24 degrees, 37 minutes, and 11.92 seconds, which to the nearest second is 12 seconds. The measure of angle 𝐵𝐴𝑀 is therefore equal to 24 degrees, 37 minutes, and 12 seconds.
As angle 𝐵𝐴𝐷 is double this, we can multiply our answer by two. This is equal to 49 degrees, 14 minutes, and 24 seconds. This is the measure of angle 𝐵𝐴𝐷 to the nearest second.
