### Video Transcript

π΄π΅πΆπ· is a quadrilateral where the measure of angle π΄π΅πΆ is 90 degrees, the measure of angle π΅π΄π· is 41 degrees, π΄π΅ is equal to π΄π·, which is equal to 30.9 centimeters, and π΅π· is equal to π΅πΆ. Find the area of π΄π΅πΆπ·, giving the answer to two decimal places.

We will begin by sketching the quadrilateral. We are told that angle π΄π΅πΆ is a right angle, and we are also told that angle π΅π΄π· is equal to 41 degrees. The lengths π΄π΅ and π΄π· are both equal to 30.9 centimeters. We are also told that the lengths of π΅π· and π΅πΆ are equal. This means that we have two isosceles triangles. The first one, triangle π΄π΅π·, 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 π΄π·π΅ is equal to angle π΄π΅π·. To calculate these angles, we can subtract 41 from 180 and then divide by two. This gives us 69.5. The angles π΄π·π΅ and π΄π΅π· are both equal to 69.5 degrees. Recalling that angle π΄π΅πΆ is equal to 90 degrees, we can calculate angle π·π΅πΆ 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 ππ multiplied by sin πΆ, where the angle πΆ lies between the two sides π and π. For the first triangle, we have enough information. However, in triangle π΅πΆπ·, we need to calculate the length of π΅πΆ and π΅π· first. We are told that these lengths are equal, so we will use the cosine rule on triangle one to calculate the length of π΅π·.

The cosine rule states that π squared is equal to π squared plus π squared minus two ππ multiplied by cos π΄, where angle π΄ is opposite the side length π weβre trying to calculate. This means that π΅π· 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 π΅π· 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 π΅π· and π΅πΆ 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 π΄π΅π· 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 π΅πΆπ· in the same way, where the two side lengths are π΅πΆ and π΅π· 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 π΄π΅πΆπ·. 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.