This quadrilateral appears to be a square, 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 P and Q.
Similar calculations give us the lengths of the other sides.
Side
Expression
Value
QR
sqrt((6-3)^2+(0-4)^2)
5
RS
sqrt((2-6)^2+(- 3-0)^2)
5
SP
sqrt((- 1-2)^2+(1-(- 3))^2)
5
We can add these lengths to find the perimeter.
5+5+5+5=20units
Finding the Area
Remember, although this quadrilateral appears to be a square, 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 small square in the middle and four congruentright triangles when we do so.
The area of the 1 by 1 square in the middle is 1* 1=1 units^2.
Counting grid units, we can find that the length of the legs of the right triangles are b=4 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 6 units^2. Adding the area of the square in the middle to these, we get the area of the quadrilateral.
6+6+6+6+1=25units^2
Alternative Solution
Is this quadrilateral a square?
Draw a diagonal and take a look at only half of the quadrilateral. What do we know about this triangle?
We have already seen that PQ=QR=5. We can now use the Distance Formula to find PR.
Since a^2+b^2=c^2, we can conclude that â–ł PQR is indeed a right triangle with the right angle at Q.
Similarly we can show that all angles of the quadrilateral PQRS are right angles.
Since we already know that the sides are congruent, we have now proven that PQRS is a square.
