In the given figure, 𝐵𝐶𝐷𝑌 is a rectangle and 𝐵 is a point on the straight line 𝐴𝐶. 𝐵𝐶 is equal to 405 meters, the measure of angle 𝐷𝐴𝐶 is 21 degrees, and the measure of angle 𝑌𝐴𝐶 is 59 degrees. Find the length of line segment 𝐷𝐶, giving the answer to the nearest meter.
We will begin by adding the information from the question onto our diagram. As 𝐵𝐶𝐷𝑌 is a rectangle, all the angles inside it will be equal to 90 degrees. We are told that 𝐵 is a point on the straight line 𝐴𝐶. Therefore, the angle 𝑌𝐵𝐴 is also 90 degrees. The measure of angle 𝐷𝐴𝐶 is 21 degrees. And since the measure of angle 𝑌𝐴𝐶 is 59 degrees, then the measure of angle 𝑌𝐴𝐷 is 59 degrees minus 21 degrees. This is equal to 38 degrees. Finally, we are told that 𝐵𝐶 has length 405 meters, which means that 𝑌𝐷 is also equal to 405 meters.
Since angles in a triangle sum to 180 degrees, the measure of angle 𝐴𝑌𝐵 is 31 degrees. We are asked to find the length of line segment 𝐷𝐶, which we have labeled 𝑥. There are many ways to do this. One way would be to split our shape into two triangles, 𝐴𝑌𝐷 and 𝐴𝐶𝐷. We know that one way to calculate missing lengths in triangles is using the sine rule or law of sines. This states that 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶. Applying this rule to triangle 𝐴𝑌𝐷, we have side length 𝐴𝐷 divided by sin of 121 degrees is equal to 405 divided by sin of 38 degrees. Multiplying through by the sin of 121 degrees, we can calculate the length of side 𝐴𝐷. This is equal to 563.8695 and so on. As this is not the final answer, we will not round. But we know that the length of 𝐴𝐷 is 563.8695 and so on meters.
We can now consider triangle 𝐴𝐷𝐶. We could use the sine rule once again as we know the measure of two angles together with the length of one side. However, since this is a right triangle, we can simply use the sine ratio, which states that the sin of angle 𝜃 is equal to the opposite over the hypotenuse. The sin of 21 degrees is equal to 𝑥 over 563.8695 and so on. Multiplying through by the denominator, we have 𝑥 is equal to 202.0727 and so on. We are asked to give our answer to the nearest meter. Since the first digit after the decimal point is zero, we round down. The length of 𝐷𝐶 correct to the nearest meter is 202 meters.