To find the area of a triangle given the coordinates of its vertices, we can use the formula:
[tex]\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\][/tex]
Given the vertices of the triangle:
- [tex]\( L(4, -6) \)[/tex]
- [tex]\( M(9, -6) \)[/tex]
- [tex]\( N(6, -2) \)[/tex]
Let’s assign these coordinates to the variables:
- [tex]\( (x_1, y_1) = (4, -6) \)[/tex]
- [tex]\( (x_2, y_2) = (9, -6) \)[/tex]
- [tex]\( (x_3, y_3) = (6, -2) \)[/tex]
Now, substitute these coordinates into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \left| 4((-6) - (-2)) + 9((-2) - (-6)) + 6((-6) - (-6)) \right|
\][/tex]
Simplify inside the absolute value:
[tex]\[
\text{Area} = \frac{1}{2} \left| 4(-6 + 2) + 9(-2 + 6) + 6(-6 + 6) \right|
\][/tex]
[tex]\[
= \frac{1}{2} \left| 4(-4) + 9(4) + 6(0) \right|
\][/tex]
[tex]\[
= \frac{1}{2} \left| -16 + 36 + 0 \right|
\][/tex]
Combine the terms inside the absolute value:
[tex]\[
= \frac{1}{2} \left| 20 \right|
\][/tex]
Which simplifies to:
[tex]\[
= \frac{1}{2} \times 20 = 10
\][/tex]
Thus, the area of the triangle is:
[tex]\[
10
\][/tex]
So the area of the triangle whose vertices are [tex]\( L(4, -6) \)[/tex], [tex]\( M(9, -6) \)[/tex], and [tex]\( N(6, -2) \)[/tex] is [tex]\( 10 \)[/tex].
