Question number 18. The side lengths of a certain triangle are expressed by integers 13, x, and y. If x times y is equal to 105, then what is the perimeter of this triangle? Let me draw in an example triangle, and then we will label the sides here. One side will be equal to 13, so maybe that's the longest side here, and then we have x and y. So the product of x and y is equal to 105, and then let's factor 105 to see what the factors can be. So that's 3 times 7 times 5, which is 21 times 5, which is also equal to 3 times 35, and finally equal to 7 times 15. So there are these different possibilities for x and y. Grouping them two at a time, we have 3 choose 2, or 3 choices here for x and y. So then let's look at each of them. If we have x is equal to 21, y is equal to 5, then this cannot be a triangle, because 5 plus 13 is equal to 18, and that is less than 21. So what we are checking here is the triangle inequality, and so the sum of two sides, any two sides, must exceed the length of the third side. So this choice here is not valid. Then we can say what if x is equal to 3, and y is equal to 35, and what happens is now we have 3 plus 13, and that is equal to 16, which is less than 35. So again, this is not possible, and so that leaves us with one choice here. x is equal to 7, for example, y is equal to 15, and then what happens is 13 plus 7 is equal to 20, which is greater than 15, and if we take any other two sides, 7 plus 15 is equal to 22, and that is greater than 13. So that works, and we have found our dimensions of this triangle, so then the perimeter is the sum, 7 plus 15 plus 13, and that comes out to 35. So the perimeter of this triangle we are looking for is 35, and that is answer A.

triangle inequality

triangle perimeter

side lengths

factor pairs

valid triangle