Given that 𝐴𝐵𝐶𝐷 is a square whose perimeter is 232 centimeters, find the lengths of line segment 𝐴𝐵 and line segment 𝐶𝐸. Then, find the area of triangle 𝐸𝐷𝐵.
So here we have our square, which has a perimeter of 232 centimeters. We can recall that the perimeter of a shape is the sum of the outside lengths. So if our square had a length of 𝑥, we would find the perimeter by adding 𝑥 plus 𝑥 plus 𝑥 plus 𝑥 or working out four times 𝑥. To find 𝑥, we take our perimeter of 232 and divide by four, giving us 58 centimeters. So now, we know that the length of every edge in the square is 58 centimeters. And so, for our first question to find the length of line segment 𝐴𝐵, we can write that 𝐴𝐵 is 58 centimeters.
The next length that we need to find is that of 𝐶𝐸. We can see from our diagram that the lengths 𝐶𝐸 and 𝐸𝐵 are marked as equal, which means that we could divide our length 𝐶𝐵 of 58 centimeters by two. And since 58 divided by two is 29, then we can say that 𝐶𝐸 is 29 centimeters. Our final part of the question is to find the area of triangle 𝐸𝐵𝐷, which we can mark in pink on the diagram. To do this, we need to use the formula that the area of a triangle is half times the base times the height. And it’s important to note that the height here refers to the perpendicular height of the triangle.
So working out the area of triangle 𝐸𝐷𝐵 using the formula, we can take 𝐸𝐵 to be the base of the triangle, which has a length of 29 centimeters. Our perpendicular height will be equivalent to line 𝐶𝐷, which is 58. And so we have the calculation a half times 29 times 58. We can simplify this calculation by noticing we have the even number 58, which we can easily take a half of, making a calculation of 29 times 29. So our answer is 841 square centimeters.
Therefore, our final answer for all parts is 𝐴𝐵 equals 58 centimeters. 𝐶𝐸 equals 29 centimeters. And the area of triangle 𝐸𝐷𝐵 equals 841 square centimeters.