QuestionFind the area of the triangle formed by the lines: , , and .
Studdy Solution
STEP 1
Assumptions1. The lines given are in the form y = mx + c, where m is the slope and c is the y-intercept.
. The lines intersect to form a triangle.
3. The area of a triangle is given by the formula Area =1/ * base * height
STEP 2
First, we need to find the points of intersection of the lines. Let's start with the intersection of the first and second lines. We can find this by setting the equations equal to each other and solving for x.
STEP 3
olve the equation for x.
STEP 4
Substitute x =8 into the first equation to find the corresponding y-coordinate.
So, the first point of intersection is (8, -1).
STEP 5
Next, find the intersection of the first and third lines.
STEP 6
olve the equation for x.
STEP 7
Substitute x =4 into the first equation to find the corresponding y-coordinate.
So, the second point of intersection is (4,7).
STEP 8
Finally, find the intersection of the second and third lines.
STEP 9
olve the equation for x.
STEP 10
Substitute x =0 into the second equation to find the corresponding y-coordinate.
So, the third point of intersection is (0,3).
STEP 11
Now that we have the three points of intersection, we can find the lengths of the sides of the triangle using the distance formula.
d = \sqrt{(x - x)^ + (y - y)^}Let's start with the distance between the first and second points.
STEP 12
Calculate the distance between the first and second points.
STEP 13
implify the equation.
STEP 14
Next, calculate the distance between the first and third points.
STEP 15
implify the equation.
STEP 16
Finally, calculate the distance between the second and third points.
STEP 17
implify the equation.
STEP 18
Now that we have the lengths of the sides, we can find the area of the triangle. Since this is a right triangle (as can be seen from the slopes of the lines), we can use the formula for the area of a right triangle Area =/2 * base * height.
STEP 19
Plug in the values for the base and the height into the formula.
STEP 20
Calculate the area of the triangle.
The area of the triangle formed by the intersecting lines is square units.
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