Exercise 8 Page 329

In this exercise we are asked to find the measure of $JQ.$ Let's find this segment on the given diagram.

From the diagram we can see that $\overline{QK},$ $\overline{QM},$ and $\overline{QP}$ are perpendicular to the sides of the triangle. This means that these segments are the distances from $Q$ to the sides of $△JLN.$ Also, we are told that $Q$ is the incenter of $△JLN.$ Thus, we can use the Incenter Theorem.

The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle.

According to the theorem, the incenter $Q$ is equidistant from the sides of the triangle $△JLN.$ This means that distances $QK,$ $QM,$ and $QP$ are the same. We also know that the measure of $QM$ is $9.$ Therefore, the measure of $QP$ is $9$ too. Now, let's consider the triangle $△JQP.$ $△JQP$ is a right triangle, because $\overline{QP}$ is perpendicular to $\overline{JP}.$ Thus, we can use the Pythagorean Theorem. In our case, the legs of the triangle are $\overline{JP}$ and $\overline{QP},$ and the hypotenuse is $\overline{JQ},$ so this theorem can be written the following way. We know the length of the two triangle's legs. Let's substitute $16.5$ for $JP$ and $9$ for $QP,$ and solve the equation we will get for $JQ.$ The measure of $JQ$ is about $18.8.$