what is the area of triangle lmn?\n3 square units\n4 square units\n6 square units\n8 square units

Explanation:

Step1: Identify base and height

Let's consider the vertical line segment from $L(1, - 1)$ to $N(1,3)$ as the height and the horizontal line - segment from $L(1,-1)$ to $M(3,2)$ as part of the base - related calculation. The length of the vertical line segment (height $h$) from $y=-1$ to $y = 3$ is $h=\vert3-( - 1)\vert=4$ units. The horizontal distance from $L(1,-1)$ to $M(3,2)$ and the perpendicular distance from $N$ to the line $LM$ are used to find the base - related value. We can also use the formula for the area of a triangle given the coordinates. Another way is to enclose the triangle in a rectangle. The rectangle that encloses $\triangle LMN$ has vertices $(1,-1),(3,-1),(3,3)$ and $(1,3)$. The area of the rectangle $A_{r}=(3 - 1)\times(3-( - 1))=8$ square units.

Step2: Subtract the areas of the two right - triangles inside the rectangle

The two right - triangles inside the rectangle: One right - triangle with vertices $(1,-1),(3,-1)$ and $(3,2)$ has area $A_{1}=\frac{1}{2}\times(3 - 1)\times(2-( - 1))=\frac{1}{2}\times2\times3 = 3$ square units. The other right - triangle with vertices $(1,-1),(1,3)$ and $(3,2)$ has area $A_{2}=\frac{1}{2}\times(3 - 1)\times(3 - 2)=1$ square unit. The area of $\triangle LMN$ is $A = A_{r}-A_{1}-A_{2}=8-3 - 1=4$ square units.

Answer:

4 square units