What is the area of the triangle whose vertices are [tex]\( L(4,-6) \)[/tex], [tex]\( M(9,-6) \)[/tex], and [tex]\( N(6,-2) \)[/tex]?

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Answer :

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].