𝐴𝐵𝐶𝐷 is a rhombus in which 𝐷𝐵 is equal to 22.4 centimeters. Determine the length of 𝐴𝐶 and the area of 𝐴𝐵𝐶𝐷.
So in this question, we’re asked to calculate two things. Firstly, the length of 𝐴𝐶, which is one of the diagonals of the rhombus. And secondly, the area of the rhombus. Let’s think about how to calculate the length of the line 𝐴𝐶, first of all. We’ll begin by focusing on the triangle that I’ve highlighted in pink, the triangle 𝐷𝐸𝐶. Now this is in fact a right-angled triangle at 𝐸, as the diagonals of a rhombus intersect at right angles. We know the length of the hypotenuse of this triangle. It’s the side length of the rhombus, 21.2 centimeters. We’re also told in the question the length of the other diagonal 𝐷𝐵 is 22.4 centimeters. This means we can calculate the length of the side 𝐷𝐸, using the fact that the diagonals of a rhombus bisect each other. Therefore, the length of the line 𝐷𝐸, which is half of the diagonal 𝐷𝐵, 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 𝐸𝐶. And in order to find 𝐴𝐶, we just need to double it. Remember, diagonals of a rhombus bisect each other. So let’s 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 𝐸𝐶 squared plus 𝐸𝐷 squared is equal to 𝐶𝐷 squared. We can replace 𝐸𝐷 and 𝐶𝐷 with their values. And therefore, we have the equation 𝐸𝐶 squared plus 11.2 squared is equal to 21.2 squared.
We now want to solve this equation to find the length of 𝐸𝐶. Evaluating both 11.2 squared and 21.2 squared gives 𝐸𝐶 squared plus 125.44 is equal to 449.44. Subtracting 125.44 from both sides gives 𝐸𝐶 squared is equal to 324. The final step in solving for 𝐸𝐶 is to take the square root of each side of the equation which gives 𝐸𝐶 is equal to the square root of 324, which is exactly 18. So now we know the length of 𝐸𝐶. We just need to double it in order to find the length of 𝐴𝐶. The length of 𝐴𝐶 is two multiplied by 18 which is 36. The units for this, which we’ll write in later, are centimeters.
So we’ve 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 𝑑 one and 𝑑 two, then the area is equal to half of their product. We know the lengths of both diagonals of this rhombus. 𝐴𝐶, we’ve calculated, is 36 centimeters. And 𝐷𝐵, 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 𝐴𝐶 is 36 centimeters. And the area of the rhombus 𝐴𝐵𝐶𝐷 is 403.2 centimeters squared.