Exercise 28 Page 62

Before we calculate the perimeter and area, let's draw a diagram of the figure with the given coordinates.

Drawing a Diagram

Let's start by plotting the points on a coordinate plane. We can see four sides, so this is a quadrilateral.

This quadrilateral appears to be a rectangle, but we cannot assume this while we are calculating the perimeter and the area.

Finding the Perimeter

The perimeter is the sum of the length of the sides. To find the length of the sides, we can use the Distance Formula. Let's start with finding the distance between $T$ and $U.$ Similar calculations give us the lengths of the other sides.

Side	Expression	Value
$UV$	$\sqrt{(5-1{)}^2+(2-6{)}^2}$	$\sqrt{32}$
$VW$	$\sqrt{(2-5{)}^2+(\text{-}1-2{)}^2}$	$\sqrt{18}$
$WT$	$\sqrt{(\text{-}2-2{)}^2+(3-(\text{-}1){)}^2}$	$\sqrt{32}$

We can add these lengths to find the perimeter.

Finding the Area

Remember, although this quadrilateral appears to be a rectangle, we cannot assume this. Instead, we can draw horizontal and vertical segments through the vertices to break this quadrilateral into smaller pieces. We can see a parallelogram in the middle and four congruent right triangles when we do so.

The area of the $6$ by $1$ parallelogram in the middle is $6\times 1=6{\text{ units}}^2.$ Counting grid units, we can find that the length of the legs of the right triangles are $b=3$ and $h=3.$ Now, we can use the formula for area of a triangle to find the area of any one of these. The area of the triangles is $4.5{\text{ units}}^2.$ Adding the area of the parallelogram in the middle to the area of all four of these triangles, we get the area of the quadrilateral.