Video: AQA GCSE Mathematics Foundation Tier Pack 1 β€’ Paper 1 β€’ Question 11

The perimeter of the equilateral triangle 𝐴 is 12 cm. The perimeter of the scalene triangle 𝐡 is 9.5 cm. Triangle 𝐡 is cut out from triangle 𝐴, leaving the quadrilateral shown. Calculate the perimeter of the quadrilateral.

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Video Transcript

The perimeter of the equilateral triangle 𝐴 is 12 centimetres. The perimeter of the scalene triangle 𝐡 is 9.5 centimetres. Triangle 𝐡 is cut out from triangle 𝐴, leaving the quadrilateral shown. Calculate the perimeter of the quadrilateral.

The first thing we need to notice is that these figures are not drawn accurately. Now let’s see what we’re given. Triangle 𝐴 is an equilateral triangle. And that means that, in triangle 𝐴, all three sides are the same length.

The perimeter of any triangle is found by adding its side one length plus its side two length plus its side three length. And in an equilateral triangle, because all three sides are the same length, we can find its perimeter by multiplying one of the side lengths by three. This also means that, for equilateral triangles, if we know their perimeter, we can find the side lengths by dividing the perimeter by three.

The perimeter of triangle 𝐴 is 12 centimetres. 12 divided by three equals four. We can say that every side of triangle 𝐴 measures four centimetres. Four plus four plus four equals 12.

Triangle 𝐡 is a little bit different. It’s a scalene triangle. And that means all three sides have different lengths. We could say that triangle 𝐡 has side lengths π‘₯, 𝑦, and 𝑧. We know that the perimeter of triangle 𝐡 is nine and a half. And that means π‘₯ plus 𝑦 plus 𝑧 must equal nine and a half.

But we know something else about triangle 𝐡. If triangle 𝐡 was cut out of triangle 𝐴 to leave this quadrilateral, this space represents triangle 𝐡. Looking at this, we can identify one of the side lengths of triangle 𝐡. Since it was cut out of triangle 𝐴, one of its sides measures four centimetres. If the 𝑧 side measures four centimetres, then nine and a half equals π‘₯ plus 𝑦 plus four.

If we take four away from nine and a half, we’ll find the length of side π‘₯ plus side 𝑦 together. π‘₯ plus 𝑦 equals five and half centimetres. We don’t know what π‘₯ or 𝑦 is, but we do know that, together, they must add up to five and a half.

Looking at this quadrilateral, we know that it was cut out from triangle 𝐴. And that means we know that the other two sides of this quadrilateral measure four centimetres. The other two sides in the quadrilateral are side length π‘₯ and 𝑦 from triangle 𝐡. The perimeter of this quadrilateral is found by adding all four sides together, side one plus side two plus side three plus side four.

Adding up all the sides would look like this: four plus four plus π‘₯ plus 𝑦. And we know what π‘₯ plus 𝑦 equals. Together, π‘₯ plus 𝑦 equals 5.5. We can substitute 5.5 in in place of π‘₯ plus 𝑦. The perimeter of this quadrilateral equals four plus four plus five and a half. Four plus four plus five and a half equals 13.5.

All of our measurements in this problem are given in centimetres. So we can say that the perimeter of this quadrilateral equals 13.5 or 13 and a half centimetres.

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