The perimeter of one of two similar triangles is 31.5 centimeters and the side lengths of the other are nine centimeters, two centimeters, and 10 centimeters. Find the length of the longest side of the first triangle rounded to two decimal places.
So we’ve been given different pieces of information about these two triangles. In, one we just know the perimeter, whereas in the second we know the individual side lengths. We were also told that the two triangles are similar, which is a really key piece of information here. Our objective is to find the length of the longest side in the first triangle.
In order to answer this question, we need to recall the proportional perimeters theorem. It tells us that if two polygons are similar, then the perimeters are proportional to the scale factor between them. We know the perimeter of the first triangle, which we’ll refer to as 𝑃 one. It’s 31.5 centimeters. And we can find the perimeter of the second triangle by summing it’s three sides. It’s 21 centimeters.
So if we look at the ratio of these two perimeters, we can then calculate the scale factor between the two triangles. This will then enable us to calculate the length of corresponding sides between the two triangles. So the ratio between the two perimeters, 𝑃 one divided by 𝑃 two, is 31.5 divided by 21. And this simplifies to three over two.
What this means then is that not only is the perimeter of the large triangle three over two or one and a half times as big as the perimeter of the smaller triangle, but the individual sides are also in the same ratio. So the longest side of the larger triangle is three over two or one and a half times as long as the longest side of the smaller triangle.
So in order to calculate the length of the longest side in the first triangle, which we’ll refer to as 𝑥 centimeters, we need to multiply the longest side of the shorter triangle, which is 10, by three over two. So we have that 𝑥 is equal to 10 multiplied by three over two, which is equal to 15. The question actually asked this value rounded to two decimal places, and it’s an exact value. So we just need to add two zeros after the decimal point.
So we have our answer to the problem. The longest side of the first triangle, to two decimal places, is 15.00 centimeters. A sensible check, which you could perform yourself, is to calculate the lengths of the other sides by multiplying the nine-centimeters and the two-centimeter side by this scale factor. You could then check that the perimeter when you add your three values together is indeed 31.5 centimeters.