The diagram shows a triangle with a quadrilateral cut out of it. All the lengths given in the diagram are in centimetres. The outer perimeter of the shape is twice the inner perimeter of the shape. Calculate the perimeter of the quadrilateral.
So in the diagram, we can see that we have this triangle and then we have the quadrilateral — that’s a four-sided shape — cut out of it. We’re asked to calculate the perimeter of the quadrilateral. And we can see that all the lengths in the diagram have been given in terms of the letter 𝑥.
In order to be able to calculate the perimeter of the quadrilateral, we first need to work out the value of 𝑥. Now, we’re told some information about the outer and inner perimeters of this shape. So let’s begin by writing down expressions for each of these.
Remember the perimeter of a shape is the total distance around its edges. So the outer perimeter of this shape is the sum of the lengths of the sides of the triangle. So that’s three 𝑥 plus five plus six 𝑥 minus two plus four 𝑥 plus eight.
Now, we can simplify this expression by collecting like terms. Looking at the 𝑥s first of all, we have three 𝑥 plus six 𝑥 which makes nine 𝑥 and then plus four 𝑥. So that makes 13𝑥. Looking at the numbers, we have positive five minus two, so that’s three, and then plus eight. So that takes us to 11. So we have a simplified expression for the outer perimeter of the shape.
Now, let’s consider the inner perimeter. This is the sum of the lengths of all of the sides of the quadrilateral. So we have three 𝑥 plus one plus two 𝑥 minus one plus 𝑥 plus two 𝑥 plus one. This can again be simplified.
Looking at the 𝑥s first, we have three 𝑥 plus two 𝑥 which is five 𝑥 plus another 𝑥 which is six 𝑥 and then plus two more 𝑥 which is eight 𝑥. Looking at the numbers, we have plus one minus one, so they cancel each other out, and then just plus one. So our expression for the inner perimeter simplifies to eight 𝑥 plus one.
Now, we’re told that the outer perimeter is twice the inner perimeter. So we can use our expressions to form an equation. 13𝑥 plus 11 is equal to two multiplied by eight 𝑥 plus one. We can now solve this equation to find the value of 𝑥.
First, we expand the bracket on the right of the equation. Two multiplied by eight 𝑥 gives 16𝑥 and two multiplied by one gives two. So now, we have 13𝑥 plus 11 equals 16𝑥 plus two. We currently have terms involving 𝑥 on both sides of the equation and we want to group them on the same side.
The side which has the larger number of 𝑥 is the right. So what we’ll do is group the 𝑥 terms on this side. To do this, we need to subtract 13𝑥 from the left of the equation but whatever we do to one side, we must also do to the other side to keep the equation balanced. On the left, we’re just left with 11 and on the right 16𝑥 minus 13𝑥 gives three 𝑥. So we now have three 𝑥 plus two.
Next, we want to eliminate the plus two on the right of the equation. So we need to subtract two. But again, whatever we do to one side of the equation, we must do to the other. So now, we have 11 minus two which is nine on the left and just three 𝑥 on the right.
The final step is to divide both sides of this equation by three as we currently have three 𝑥 and we want to know what 𝑥 or one 𝑥 is equal to. Nine divided by three is three. So we’ve solved the equation. And we have three equals 𝑥 or 𝑥 equals three.
Now, remember our objective in this question was to calculate the perimeter of the quadrilateral. Our expression for the inner perimeter which was the perimeter of the quadrilateral was eight 𝑥 plus one.
So we can now substitute this value of 𝑥 into this expression in order to calculate the perimeter. And it gives eight multiplied by three plus one. Eight multiplied by three is 24 and adding one gives 25.
The question tells us that all the lengths in the diagram are given in centimetres and its perimeter is also a measure of length; it will have the same units.
So the perimeter of the quadrilateral is 25 centimetres.