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".
