The angle at $Q$ is formed by sides $PQ$ and $QR$. According to the Angle Bisector Theorem, how is $PS:SR$ related to $PQ:QR$?

Once you have $\frac{PS}{SR}$ in terms of known numbers, try writing $PS$ and $SR$ as $6k$ and $5k$ (or a similar pair) so their ratio matches. How can you then use $PS + SR = 12$ to find $k$?

After you get an equation like $11k = 12$, solve for $k$ and then substitute back to get $PS$. Be careful to compute the final fraction accurately.

In Desmos, type the expression

Here, $x$ represents $PS$, and $12 - x$ represents $SR$. This expression will be zero when the ratio $\frac{PS}{SR}$ equals $\frac{18}{15}$ as required by the Angle Bisector Theorem.

Look at the graph of the expression and find the x-intercept (where the graph crosses the x-axis). The x-coordinate of that point is the value of $x$ that makes the expression zero, which is the length of $PS$.

We are given triangle $PQR$ with side lengths:

• $PQ = 18$
• $QR = 15$
• $PR = 12$

Point $S$ lies on side $PR$, and $QS$ is an angle bisector of $\angle Q$. That means $S$ splits side $PR$ into two segments:

• $PS$ (the piece we want)
• $SR$

We also know:

The Angle Bisector Theorem says: if a segment from a vertex of a triangle bisects the angle at that vertex, then it divides the opposite side into segments that are proportional to the two sides forming the angle.

Here, $QS$ bisects $\angle Q$, which is between sides $PQ$ and $QR$. So

Substitute the known side lengths $PQ = 18$ and $QR = 15$:

Simplify the fraction:

So

From

we can let

• $PS = 6k$
• $SR = 5k$

for some positive number $k$. This keeps the ratio $PS:SR = 6:5$.

We also know their sum is 12:

Substitute $PS = 6k$ and $SR = 5k$:

So

From

solve for $k$:

Now recall $PS = 6k$. Substitute $k$:

So the length of $PS$ is