Question

In triangle PQR, the length of PQ is 5 inches and the length of PR is 9 inches. Give all the possible whole-number lengths of side QR.

Answer 1

Answer:

$4$

5, 6, 7, 8, 9, 10, 11, 12, and 13.

Step-by-step explanation:

We can use the triangle inequality. By the triangle inequality, the sum of any two sides of a triangle must be greater than the remaining side.

In other words, the sum of the two shorter sides must be greater than the longest side.

We know that PQ is 5 and PR is 9.

For the first case, let's say that neither PQ nor PR is the longest side. Instead, QR is the longest side. Then their sum must be greater than QR. Thus:

$5+9>QR\Rightarrow QR$

However, for the second case, let's say that QR is not the longest side. In this case, PR will be the longest side. Therefore, QR plus PQ must be greater than PR:

$QR+5>9\Rightarrow QR>4$

Thus, we have a compound inequality:

$4$

Therefore, the possible whole-number lengths of QR are:

5, 6, 7, 8, 9, 10, 11, 12, and 13.