We are given a triangle $PQR$ with several geometric conditions, and we are tasked with finding the area of triangle $PYR$. Let's break down the problem step by step and use the given information to solve it.

Step 1: Analyze the triangle and given points

• $M$ is the midpoint of $\overline{PQ}$, which implies that $PM = MQ = \frac{PQ}{2} = 14$.
• $X$ is a point on $\overline{QR}$, and $\overline{PX}$ bisects $\angle QPR$. This tells us that $\overline{PX}$ is the angle bisector of $\angle QPR$.
• The perpendicular bisector of $\overline{PQ}$ intersects $\overline{PX}$ at point $Y$, and it is given that $MY = 5$.

Step 2: Use geometric properties

We know that:

• The perpendicular bisector of $\overline{PQ}$ implies that point $Y$ is equidistant from $P$ and $Q$, because the perpendicular bisector of a segment is the locus of points equidistant from the endpoints of the segment.
• By the Angle Bisector Theorem, since $\overline{PX}$ is the angle bisector of $\angle QPR$, the ratio $\frac{QX}{XR}$ is equal to the ratio of the adjacent sides $\frac{PQ}{PR} = \frac{28}{16} = \frac{7}{4}$.

Step 3: Setup coordinate geometry

To solve for the area of triangle $PYR$, we can place the triangle on a coordinate plane. Let the coordinates of point $P$ be $(0, 0)$, point $Q$ be $(28, 0)$, and point $R$ be $(0, 16)$.

• Since $M$ is the midpoint of $\overline{PQ}$, its coordinates are $M = \left( \frac{28 + 0}{2}, \frac{0 + 0}{2} \right) = (14, 0)$.

Now, since $Y$ lies on the perpendicular bisector of $\overline{PQ}$, its $x$-coordinate must be 14, the same as point $M$. The distance from $M$ to $Y$ is given as 5, so the $y$-coordinate of $Y$ must be 5 units above or below $M$ along the perpendicular bisector. Therefore, the coordinates of $Y$ are $(14, 5)$.

Step 4: Calculate the area of triangle $PYR$

We now know the coordinates of the vertices of triangle $PYR$:

• $P = (0, 0)$
• $Y = (14, 5)$
• $R = (0, 16)$

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by the formula:

$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substituting the coordinates of $P$, $Y$, and $R$:

$\text{Area} = \frac{1}{2} \left| 0(5 - 16) + 14(16 - 0) + 0(0 - 5) \right|$ $= \frac{1}{2} \left| 0 + 14 \times 16 + 0 \right|$ $= \frac{1}{2} \times 224 = 112$

Thus, the area of triangle $PYR$ is $\boxed{112}$.

Follow-up questions

1. How would the result change if $PQ = 30$ and $PR = 20$?
2. Can you derive the coordinates of point $X$ using the Angle Bisector Theorem?
3. How does the location of $M$ influence the position of $Y$?
4. How would the problem change if $MY = 10$?
5. Can you generalize the method to any triangle with a similar configuration?

Tip

Using coordinate geometry is an effective way to solve complex geometric problems by translating them into algebraic equations.