Question

A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (2,1). Write the area of the triangle as a function of x, and determine the domain of the function.

Answer 1

1) Find the relationship betwenn y and x.

You do this trhough the slope equation.

You have three points. (0,y) , (2,1) , and (x,0)

Slope between (0,y) and (2,1) = [y - 1] / [ 0 -2] = [1 - y] / 2

Slope between (2,1) and (x,0) = [1 - 0] / [2 -x] = 1 / [2 - x]

Then, [1 - y] / 2 = 1 / [2 - x] => 1 - y = 2 / [2-x] =>

y = 1 - 2 /[2 - x] = (2 - x - 2) / (2 - x) = -x / (2 - x) = x / (x -2)

2) Now establish the formula for the area of the triangle:

(1 / 2) base*heigth = (1/2) xy

3) replace y with x / (x - 2)

area = (1/2) x [ x / (x-2)] = [x^2] / [2(x-2)] = x^2 / (2x - 4)

Answer: area = x^2 / (2x - 4)

4) Domain:

x > 2 to y be positive,

Answer: x ∈ (2, ∞)