Final answer:
To calculate the area of triangle KLM, use the formula A = (1/2) * base * height. The area of triangle KLM is equal to the area of triangle QLR due to proportional dimensions. The length of QR can be found using the Pythagorean theorem.
Step-by-step explanation:
To calculate the area of triangle KLM, we can use the formula A = (1/2) * base * height. The base of triangle KLM is equal to the side length KL, and the height is equal to the side length KM. Since KLQP is a square, KL = QL. Similarly, since MLRS is a square, KM = LR. Therefore, the area of triangle KLM is:
A = (1/2) * KL * KM = (1/2) * QL * LR
Similarly, to calculate the area of triangle QLR, we can use the same formula. The base of triangle QLR is equal to the side length LR, which is equal to KM. The height of triangle QLR is equal to the side length QR. Therefore, the area of triangle QLR is:
A = (1/2) * LR * QR = (1/2) * KM * QR
The reason why the area of triangle KLM is equal to the area of triangle QLR is because the base and height of both triangles are proportional, and the proportionality factor is 1/2.
To find the area of the pentagon KSROP, we can subtract the area of triangle KLM from the area of the larger triangle KLMQR. Since we already know the areas of triangle KLM and QLR, we can calculate the area of the pentagon as:
Area of Pentagon KSROP = Area of Triangle KLMQR - Area of Triangle KLM
Finally, to calculate the length of QR, we can use the Pythagorean theorem. In triangle QLR, QR is the hypotenuse, and LR and QL are the other two sides. Therefore, we can use the formula:
QR = sqrt(LR^2 + QL^2)