In the trapezoid 𝐴𝐵𝐶𝐷, the measure of angle 𝐵𝐴𝐻 is equal to 30 degrees. Find the area of 𝐴𝐵𝐶𝐷 in square feet. Round your answer to the nearest tenth.
We’re told in the question that the angle 𝐵𝐴𝐻 is equal to 30 degrees. In order to calculate the area of the trapezoid or trapezium, we firstly need to calculate the length of 𝐴𝐻, labeled 𝑥 on the diagram. We can do this using the trig ratios in right-angled triangles and our knowledge of special trig angles. In the triangle 𝐴𝐵𝐻, the length 𝐴𝐵 is the hypotenuse as this is opposite the right angle. The length 𝐵𝐻 is the opposite as it is opposite the 30-degree angle. Finally, the length 𝐴𝐻 is the adjacent.
We can calculate the length 𝐵𝐻 using the fact that the trapezoid is regular. The length 𝐵𝐻 will be equal to the length 𝐸𝐶. As 𝐻𝐸 will be equal to six inches, we can calculate the length of 𝐵𝐻 by subtracting six from 20 and halving our answer. This means that 𝐵𝐻 is equal to seven inches. We’re dealing with the opposite, an adjacent side of the right-angled triangle. This means we will use the tangent ratio. Tan 𝜃 is equal to the opposite over the adjacent.
Substituting in our values, we have tan 30 is equal to seven over 𝑥. Tan of 30 degrees is one of our special trig angles. It is equal to root three over three. This can also be written as one over root three, which we’ll use in this case. One over root three is equal to seven over 𝑥. Cross multiplying here by multiplying both sides by 𝑥 and root three gives us 𝑥 is equal to seven root three inches. This is equal to 12.12 and so on. Therefore, the length of 𝐴𝐻 is 12.12 inches.
The area of any trapezoid or trapezium can be calculated using the formula 𝑎 plus 𝑏 divided by two multiplied by ℎ, where 𝑎 and 𝑏 are the lengths of the parallel sides and ℎ is the perpendicular distance or height between them. In our question, 𝑎 is equal to 20 inches, 𝑏 is equal to six inches, and ℎ is equal to 12.12 inches. It is not as straightforward as just substituting in these numbers, as we’re asked to give our answer in square feet. We know that 12 inches is equal to one foot. So we need to convert these values into feet first.
20 divided by 12 is equal to 1.6 recurring. Therefore, the length of 𝑎 is 1.6 recurring feet. Six divided by 12 is equal to 0.5. So the length of 𝑏 is 0.5 feet. Finally, the length of ℎ is 1.01 and so on feet. We can then calculate the area by substituting in these values. This is equal to 1.0941 and so on. We’re asked to give our answer to the nearest tenth. So we need to round to one decimal place. The deciding number is the nine in the hundredths column. If this deciding number is five or greater, we round up. The area of the trapezoid to the nearest tenth is 1.1 square feet.