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

𝐴𝐡𝐢𝐷 is a quadrilateral where π‘šβˆ π΄π΅πΆ = 90Β°, π‘šβˆ π΅π΄π· = 41Β°, 𝐴𝐡 = 𝐴𝐷 = 30.9 cm, and 𝐡𝐷 = 𝐡𝐢. Find the area of 𝐴𝐡𝐢𝐷 giving the answer to two decimal places.

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Video Transcript

𝐴𝐡𝐢𝐷 is a quadrilateral where the measure of angle 𝐴𝐡𝐢 is 90 degrees, the measure of angle 𝐡𝐴𝐷 is 41 degrees, 𝐴𝐡 is equal to 𝐴𝐷, which is equal to 30.9 centimeters, and 𝐡𝐷 is equal to 𝐡𝐢. Find the area of 𝐴𝐡𝐢𝐷, giving the answer to two decimal places.

We will begin by sketching the quadrilateral. We are told that angle 𝐴𝐡𝐢 is a right angle, and we are also told that angle 𝐡𝐴𝐷 is equal to 41 degrees. The lengths 𝐴𝐡 and 𝐴𝐷 are both equal to 30.9 centimeters. We are also told that the lengths of 𝐡𝐷 and 𝐡𝐢 are equal. This means that we have two isosceles triangles. The first one, triangle 𝐴𝐡𝐷, 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 𝐴𝐷𝐡 is equal to angle 𝐴𝐡𝐷. To calculate these angles, we can subtract 41 from 180 and then divide by two. This gives us 69.5. The angles 𝐴𝐷𝐡 and 𝐴𝐡𝐷 are both equal to 69.5 degrees. Recalling that angle 𝐴𝐡𝐢 is equal to 90 degrees, we can calculate angle 𝐷𝐡𝐢 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 π‘Žπ‘ multiplied by sin 𝐢, where the angle 𝐢 lies between the two sides π‘Ž and 𝑏. For the first triangle, we have enough information. However, in triangle 𝐡𝐢𝐷, we need to calculate the length of 𝐡𝐢 and 𝐡𝐷 first. We are told that these lengths are equal, so we will use the cosine rule on triangle one to calculate the length of 𝐡𝐷.

The cosine rule states that π‘Ž squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 multiplied by cos 𝐴, where angle 𝐴 is opposite the side length π‘Ž we’re trying to calculate. This means that 𝐡𝐷 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 𝐡𝐷 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 𝐡𝐷 and 𝐡𝐢 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 𝐴𝐡𝐷 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 𝐡𝐢𝐷 in the same way, where the two side lengths are 𝐡𝐢 and 𝐡𝐷 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 𝐴𝐡𝐢𝐷. 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.

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