Answer:
The area of a right-angled triangle is given by the formula A = (1/2)bh, where b is the length of the base and h is the length of the height. We are given that the area of the triangle is 42 cm^2, so we can write:
42 = (1/2)(x+3)(x+8)
Simplifying the right-hand side, we get:
84 = (x+3)(x+8)
Expanding the brackets, we get:
84 = x^2 + 11x + 24
Rearranging, we get:
x^2 + 11x - 60 = 0
We can factorize the left-hand side to get:
(x + 15)(x - 4) = 0
So either x + 15 = 0 or x - 4 = 0. Solving for x in each case, we get:
x = -15 or x = 4
Since the lengths of the base and height of the triangle must be positive, we can disregard the solution x = -15. Therefore, the length of the base is:
x + 3 = 4 + 3 = 7 cm
And the length of the height is:
x + 8 = 4 + 8 = 12 cm
So the base of the triangle is 7 cm and the height is 12 cm.