### Video Transcript

π΄π΅πΆπ· is a rhombus in which π·π΅ is equal to 22.4 centimeters. Determine the length of π΄πΆ and the area of π΄π΅πΆπ·.

So in this question, weβre asked to calculate two things. Firstly, the length of π΄πΆ, which is one of the diagonals of the rhombus. And secondly, the area of the rhombus. Letβs think about how to calculate the length of the line π΄πΆ, first of all. Weβll begin by focusing on the triangle that Iβve highlighted in pink, the triangle π·πΈπΆ. Now this is in fact a right-angled triangle at πΈ, as the diagonals of a rhombus intersect at right angles. We know the length of the hypotenuse of this triangle. Itβs the side length of the rhombus, 21.2 centimeters. Weβre also told in the question the length of the other diagonal π·π΅ is 22.4 centimeters. This means we can calculate the length of the side π·πΈ, using the fact that the diagonals of a rhombus bisect each other. Therefore, the length of the line π·πΈ, which is half of the diagonal π·π΅, is 11.2 centimeters.

We now have a right-angled triangle in which we know the length of two of the sides. Therefore, we can apply the Pythagorean theorem to calculate the length of the third side. This will give us the length of the line segment πΈπΆ. And in order to find π΄πΆ, we just need to double it. Remember, diagonals of a rhombus bisect each other. So letβs recall the Pythagorean theorem. It tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In our triangle, this means that πΈπΆ squared plus πΈπ· squared is equal to πΆπ· squared. We can replace πΈπ· and πΆπ· with their values. And therefore, we have the equation πΈπΆ squared plus 11.2 squared is equal to 21.2 squared.

We now want to solve this equation to find the length of πΈπΆ. Evaluating both 11.2 squared and 21.2 squared gives πΈπΆ squared plus 125.44 is equal to 449.44. Subtracting 125.44 from both sides gives πΈπΆ squared is equal to 324. The final step in solving for πΈπΆ is to take the square root of each side of the equation which gives πΈπΆ is equal to the square root of 324, which is exactly 18. So now we know the length of πΈπΆ. We just need to double it in order to find the length of π΄πΆ. The length of π΄πΆ is two multiplied by 18 which is 36. The units for this, which weβll write in later, are centimeters.

So weβve answered the first part of the question. And now we need to find the area of this rhombus. So we need to recall the formula for doing so. The area of a rhombus can be calculated using its diagonals. If the diagonals of the rhombus are π one and π two, then the area is equal to half of their product. We know the lengths of both diagonals of this rhombus. π΄πΆ, weβve calculated, is 36 centimeters. And π·π΅, remember, was given in the question, 22.4 centimeters. Therefore, our calculation for the area is one-half multiplied by 36 multiplied by 22.4. Evaluating this gives 403.2.

So we have our answer to the problem, now with the appropriate units for each part. The length of the diagonal π΄πΆ is 36 centimeters. And the area of the rhombus π΄π΅πΆπ· is 403.2 centimeters squared.