Question

find the area of the polygon with the given vertices. k(-3,4), l(1,4), m(-4,-2), n(0,-2)

Explanation:

Step1: Find length of base

The points $K(-3,4)$ and $L(1,4)$ have the same $y -$coordinate. The length of $KL$ (base) is calculated using the distance formula for points with the same $y -$value, which is $d=\vert x_2 - x_1\vert$. So, $KL=\vert1-(-3)\vert=\vert1 + 3\vert = 4$.

Step2: Find length of height

The points $K(-3,4)$ and $M(-4,-2)$. The height $h$ between the parallel lines (since the $y -$values of $K,L$ are the same and $y -$values of $M,N$ are the same) is the vertical distance. Using the distance formula for points with different $y -$values $d=\vert y_2 - y_1\vert$, we take a point from the top - row ($K$ or $L$) and a point from the bottom - row ($M$ or $N$). Let's take $K(-3,4)$ and $M(-4,-2)$. The height $h=\vert4-(-2)\vert=\vert4 + 2\vert=6$.

Step3: Calculate area of polygon

The polygon $KLMN$ is a trapezoid. The area formula of a trapezoid is $A=\frac{1}{2}(b_1 + b_2)h$. Here, we can also consider it as a combination of two non - overlapping triangles or use the fact that we can split it into rectangles and triangles. Another way is to use the formula for the area of a polygon given vertices. But a simple way is to note that the polygon can be thought of as a rectangle - like shape. The area $A =$ (base) $\times$ (height). Substituting the values of base $b = 4$ and height $h=6$, we get $A=4\times6 = 24$.

Answer:

24