Video Transcript
A rectangle has vertices at the points ๐ด, ๐ต, ๐ถ, and ๐ท with coordinates one, one; four, two; six, negative four; and three, negative five, respectively. Firstly, work out the perimeter of the rectangle ๐ด๐ต๐ถ๐ท. Give your solution to two decimal places. Secondly, work out the area of the rectangle ๐ด๐ต๐ถ๐ท.
So weโre given the coordinates of the four vertices of a rectangle. And weโre 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 ๐ฟ represent the length and ๐ 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 donโt 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. Weโll 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 ๐ฅ one, ๐ฆ one and ๐ฅ two, ๐ฆ two. The distance between these points is the square root of ๐ฅ two minus ๐ฅ one all squared plus ๐ฆ two minus ๐ฆ one all squared, which is just an application of the Pythagorean theorem.
Weโll begin by calculating the length of the side ๐ด๐ต. For the ๐ฅ values, this gives four minus one all squared and for ๐ฆ 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. Weโll leave the length of ๐ด๐ต as a surd for now.
Now we need to calculate the length of one of the adjacent sides. So weโll choose the side ๐ถ๐ต. 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. Itโs 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 weโre able to calculate its perimeter. Remember, the perimeter is twice the length plus twice the width. So itโs 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, itโs 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 letโs 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. Theyโre 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 โ itโs an exact value โ itโs 20.
