A rectangle has vertices at the points 𝐴, 𝐵, 𝐶, and 𝐷 with coordinates one, one; four, two; six, negative four; and three, negative five, respectively. Firstly, work out the perimeter of the rectangle 𝐴𝐵𝐶𝐷. Give your solution to two decimal places. Secondly, work out the area of the rectangle 𝐴𝐵𝐶𝐷.
So we’re given the coordinates of the four vertices of a rectangle. And we’re asked us to calculate both its perimeter and its area. We start with the perimeter.
The perimeter of a rectangle is found by summing together the lengths of its four sides. If we let 𝐿 represent the length and 𝑊 the width of the rectangle, then the perimeter is calculated by multiplying the length by two and the width by two and adding them together.
So we don’t need to actually calculate the lengths of all of the sides of the rectangle individually because, of course, opposite sides are the same length. So we just need to calculate two adjacent sides. We’ll do this using the distance formula.
The distance formula tells us how to calculate the distance between two points on a coordinate grid with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. The distance between these points is the square root of 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared, which is just an application of the Pythagorean theorem.
We’ll begin by calculating the length of the side 𝐴𝐵. For the 𝑥 values, this gives four minus one all squared and for 𝑦 two minus one all squared. This is equal to the square root of three squared plus one squared. Three squared is nine and one squared is one. So we have the square root of 10. We’ll leave the length of 𝐴𝐵 as a surd for now.
Now we need to calculate the length of one of the adjacent sides. So we’ll choose the side 𝐶𝐵. The calculation here is the square root of six minus four all squared plus negative four minus two all squared. This gives the square root of two squared plus negative six squared. Two squared is four and negative six squared is 36. So we have the square root of 40.
This surd can be simplified because 40 has a square factor. It’s equal to four times 10. So we have the square root of four multiplied by the square root of 10. And this simplifies to two root 10.
So now we know about the length and the width of the rectangle. So we’re able to calculate its perimeter. Remember, the perimeter is twice the length plus twice the width. So it’s two multiplied by root 10 plus two lots of two root 10. This gives two root 10 plus four root 10, which simplifies to six root 10.
Remember, the question asked us for a solution not as a surd but to two decimal places. So we now need to use a calculator to evaluate this. As a decimal, it’s 18.97366. And if you round it to two decimal places, we have 18.97.
So we found the perimeter of the rectangle by using the distance formula to calculate the lengths of two of the adjacent sides. Now let’s focus on calculating the area. The area of a rectangle is found by multiplying its length by its width. We already know both of these. They’re root 10 and two root 10. So our calculation for the area is root 10 multiplied by two root 10. Now root 10 multiplied by root 10 just gives 10. So we have two multiplied by 10, which is equal to 20.
So our final answer to the problem then: the perimeter of this rectangle to two decimal places is 18.97. And the area of the rectangle — it’s an exact value — it’s 20.