Finding the Area of a Triangle with Given Sides

Given the sides of a triangle along the lines x = 0, y = 0, and 4x - 5y = 20, we can find the area of the triangle using the following steps:

Step 1: Find the coordinates of the vertices of the triangle.
Since the triangle has sides along the lines x = 0 and y = 0, two of its vertices are at the origin (0,0). To find the third vertex, we solve the equation 4x - 5y = 20 for y when x = 0 and for x when y = 0:
When x = 0: -5y = 20, y = -4
When y = 0: 4x = 20, x = 5
So the third vertex is (5,0).

Step 2: Find the length of each side of the triangle.
We can use the distance formula to find the length of each side of the triangle. The distance formula is:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we get:

Side 1: distance from (0,0) to (5,0)
distance = sqrt((5 - 0)^2 + (0 - 0)^2) = 5

Side 2: distance from (0,0) to (0,-4)
distance = sqrt((0 - 0)^2 + (-4 - 0)^2) = 4

Side 3: distance from (5,0) to (0,-4)
distance = sqrt((0 - 5)^2 + (-4 - 0)^2) = sqrt(41)

Step 3: Use Heron's formula to find the area of the triangle.
Heron's formula is a formula for finding the area of a triangle in terms of its side lengths. The formula is:

area = sqrt(s(s-a)(s-b)(s-c))

where a, b, and c are the side lengths of the triangle and s is the semiperimeter, which is half the perimeter of the triangle:

s = (a + b + c)/2

Using the side lengths we found in step 2, we get:

s = (5 + 4 + sqrt(41))/2 = (9 + sqrt(41))/2

area = sqrt((9 + sqrt(41))/2(9 + sqrt(41))/2-5)(9 + sqrt(41))/2-4)((9 + sqrt(41))/2-sqrt(41))) = sqrt(20) = 2sqrt(5)

Therefore, the area of the triangle is 2sqrt(5), which is approximately equal to 4.47. The correct option is (c) 10.