Area of triangle = (1/2)(base)(height)
Draw a diagram.
Let (c,0) be the x-intercept of the line y = (-x/2) + k
So, 0 = -c/2 + k Therefore, c = 2k
So, the base of the triangle is the line segment with endpoints (0,0) and (c,0).
Thus, the length of the base is 2k.
The lines y = 2x and y = -x/2 + k intersect when 2x = -x/2 + k
(5/2)x = k
x = (2/5)k
y = 2x = (4/5)k
= height of triangle
Area of triangle = (1/2)(base)(height) = (1/2)(2k)(4/5)k = 80
(4/5)k2 = 80
k2 = 100
k = ±10
But, by assumption, k > 0. So, k = 10.