The shape 𝑆 is a square. A rectangle is cut from 𝑆 to leave the shape 𝐶. Which of these statements is true? Tick a box. The perimeter of 𝑆 is longer than the perimeter of 𝐶. The perimeter of 𝑆 is the same as the perimeter of 𝐶. The perimeter of 𝑆 is shorter than the perimeter of 𝐶. Or it is not possible to tell which perimeter is longer.
So if we’re looking at the perimeter of 𝑆, the perimeter of 𝑆 is the distance around the outside of the shape 𝑆. And we know that each of the sides of our square is going to be the same length because a square has four equal sides. Well, now what we need to do is compare the perimeter of 𝑆 to the perimeter of 𝐶. Well, let’s see if we can compare it side by side.
So if we look at 𝑆 and 𝐶, they share this side. So we know that this part is exactly the same length. They also share the top side because this is the same in both 𝑆 and 𝐶. The bottom side is also shared. So now there’s just one side left from the perimeter of 𝑆.
Well, now if we want to look at the final side, we need to consider three parts from 𝐶. We have the top part or the top arm. Then we have the bit where it goes in. Then we have the bit at the bottom. And if we look at the vertical lengths of each of these, if we pull them across and stack them one on top of the other, we can see that it’ll make a vertically straight line. So therefore, this vertically straight line is the same height as one side of our square, cause it’s the same length as the other side on 𝐶 on the left-hand side.
So therefore, we’ve now covered all of the sides of our square. So it must, therefore, have at this point the same perimeter as our square 𝑆. But have we taken all the sides of the new shape 𝐶? Well, no, because if we’re looking for the perimeter of 𝐶, it means walking all the way around the outside of the shape. Well, in that case, there are still two sides that we haven’t accounted for. So therefore, we can say the perimeter of 𝐶 is equal to the perimeter of 𝑆 plus two blue lines, because I’ve coloured those two lines in blue.
Okay, so now let’s take a look at our statements. So which one of these is going to be correct? Well, we can see that the perimeter of 𝑆 is shorter than the perimeter of 𝐶, because the perimeter of 𝐶 is equal to the perimeter of 𝑆 plus two blue lines. So therefore, the third statement is the correct statement, because this says that the perimeter of 𝑆 is shorter than the perimeter of 𝐶. And we’ve proved that by showing that the perimeter of 𝐶 is greater or longer than the perimeter of 𝑆 because it is the perimeter of 𝑆 plus two extra lines.