Question Video: Calculating the Area of a Quadrilateral using Trigonometry Mathematics

Video Transcript

𝐴𝐵𝐶𝐷 is a quadrilateral where the measure of angle 𝐴𝐵𝐶 is 90 degrees, the measure of angle 𝐵𝐴𝐷 is 41 degrees, 𝐴𝐵 is equal to 𝐴𝐷, which is equal to 30.9 centimeters, and 𝐵𝐷 is equal to 𝐵𝐶. Find the area of 𝐴𝐵𝐶𝐷, giving the answer to two decimal places.

We will begin by sketching the quadrilateral. We are told that angle 𝐴𝐵𝐶 is a right angle, and we are also told that angle 𝐵𝐴𝐷 is equal to 41 degrees. The lengths 𝐴𝐵 and 𝐴𝐷 are both equal to 30.9 centimeters. We are also told that the lengths of 𝐵𝐷 and 𝐵𝐶 are equal. This means that we have two isosceles triangles. The first one, triangle 𝐴𝐵𝐷, has two sides of length 30.9 centimeters.

We know that the angles in any triangle sum to 180 degrees. And that in an isosceles triangle, two of the angles are equal. Angle 𝐴𝐷𝐵 is equal to angle 𝐴𝐵𝐷. To calculate these angles, we can subtract 41 from 180 and then divide by two. This gives us 69.5. The angles 𝐴𝐷𝐵 and 𝐴𝐵𝐷 are both equal to 69.5 degrees. Recalling that angle 𝐴𝐵𝐶 is equal to 90 degrees, we can calculate angle 𝐷𝐵𝐶 by subtracting 69.5 from 90. This is equal to 20.5 degrees.

We know that we can calculate the area of any triangle using the formula a half 𝑎𝑏 multiplied by sin 𝐶, where the angle 𝐶 lies between the two sides 𝑎 and 𝑏. For the first triangle, we have enough information. However, in triangle 𝐵𝐶𝐷, we need to calculate the length of 𝐵𝐶 and 𝐵𝐷 first. We are told that these lengths are equal, so we will use the cosine rule on triangle one to calculate the length of 𝐵𝐷.

The cosine rule states that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 multiplied by cos 𝐴, where angle 𝐴 is opposite the side length 𝑎 we’re trying to calculate. This means that 𝐵𝐷 squared is equal to 30.9 squared plus 30.9 squared minus two multiplied by 30.9 multiplied by 30.9 multiplied by cos of 41 degrees. Typing the right-hand side into our calculator gives us 468.411 and so on. We can then square root both sides of this equation so that 𝐵𝐷 is equal to 21.642 and so on.

When performing the remainder of our calculations, we will use the unrounded answer. However, we can see that 𝐵𝐷 and 𝐵𝐶 are both equal to 21.6 centimeters to one decimal place. We can now work out the areas of triangle one and triangle two. The area of triangle 𝐴𝐵𝐷 is equal to a half multiplied by 30.9 multiplied by 30.9 multiplied by sin 41 degrees. This is equal to 313.20586 and so on.

We can calculate the area of triangle 𝐵𝐶𝐷 in the same way, where the two side lengths are 𝐵𝐶 and 𝐵𝐷 and the angle we use is 20.5 degrees. This gives us an answer of 82.02058 and so on. Adding the areas of these two triangles will give us the area of the quadrilateral 𝐴𝐵𝐶𝐷. This is equal to 395.22644 and so on.

We are asked to give our answer to two decimal places. So, the deciding number is the six. Rounding up, we can conclude that the area of the quadrilateral is 395.23 square centimeters.