In the diagram, find PQ if the area of triangle PQR is 35cm2

To find the length of PQ, we need to use the formula for the area of a triangle, which is: Area = 1/2 * base * height In this case, the area is given as 35cm^2, and we know that the height is PQ. We also know that the base is QR, since the height is perpendicular to the base. So, we can write: 35 = 1/2 * QR * PQ Multiplying both sides by 2 and dividing by PQ, we get: 70/PQ = QR We don't have the value of QR, but we can use the Pythagorean theorem to find it. In triangle PQR, we have: PR^2 = PQ^2 + QR^2 We don't know the value of PR, but we can express it in terms of PQ and QR using the fact that PQR is a right-angled triangle. We have: PR = sqrt(PQ^2 + QR^2) Substituting this expression into the Pythagorean theorem, we get: PQ^2 + QR^2 = (sqrt(PQ^2 + QR^2))^2 Simplifying, we get: PQ^2 + QR^2 = PQ^2 + QR^2 This is an identity, which means it is true for all values of PQ and QR. But we can use it to eliminate QR from the equation we got earlier: 70/PQ = QR Multiplying both sides by QR, we get: 70 = PQ^2 + QR^2 Substituting QR^2 = 70 - PQ^2, we get: 70/PQ = sqrt(70 - PQ^2) Squaring both sides, we get: 4900/PQ^2 = 70 - PQ^2 Rearranging, we get: PQ^4 - 70PQ^2 + 4900 = 0 This is a quadratic equation in PQ^2, which we can solve using the quadratic formula: PQ^2 = (70 ± sqrt(70^2 - 4*1*4900))/2 PQ^2 = (70 ± 20sqrt(3))/2 We can discard the negative solution, since PQ is a length and can't be negative. So, we have: PQ^2 = 35 + 10sqrt(3) Taking the square root, we get: PQ ≈ 7.98 So, the closest option is 8cm, which corresponds to the first option, "7cm".