The area of the triangle made by the line x/a + y/b = 1 with the coordinate axis is ------how ?
check the intercepts.
when x=0, y=b
when y=0, x=a
so, you have a right triangle with legs a and b, so its area is ab/2
To find the area of the triangle, you need to determine the coordinates of the vertices of the triangle formed by the line x/a + y/b = 1 with the coordinate axes. Then you can use the formula for the area of a triangle.
The equation of the line x/a + y/b = 1 can be rearranged to solve for y in terms of x:
y/b = 1 - x/a
y = b - bx/a
To find the coordinates of the vertices, you need to identify the points where the line intersects the x-axis and the y-axis. Let's start with the x-axis:
y = 0:
0 = b - bx/a
Solving for x:
b = bx/a
x = a
The coordinate for the point of intersection with the x-axis is (a, 0).
Now, let's look at the y-axis:
x = 0:
y = b - b(0)/a
y = b
The coordinate for the point of intersection with the y-axis is (0, b).
Finally, we have the third vertex of the triangle, which is the point of intersection of the line with the coordinate axes. This point can be found by setting both x and y to zero:
0/a + 0/b = 1
0 = b
The coordinate for the third vertex is (0, 0).
Now that we have the coordinates of the three vertices, we can calculate the area of the triangle using the formula for the area of a triangle:
Area = 1/2 * |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|
Using the coordinates we found, the area of the triangle can be calculated as follows:
Area = 1/2 * |(a(0 - b) + 0(b - 0) + 0(b - 0))|
= 1/2 * |(-ab)|
= 1/2 * ab
= ab/2
Therefore, the area of the triangle formed by the line x/a + y/b = 1 with the coordinate axis is ab/2.