The problem states that triangle MNO is similar to triangle PQR, and we are asked to find the measure of side PQ.

Since the two triangles are similar, the corresponding sides of the triangles are proportional. The side lengths provided for triangle MNO are:

• $MN = 23$
• $NO = 17$

In triangle PQR, the corresponding sides are:

• $QR = 56$

Let the length of side $PQ$ be $x$.

Using the property of similar triangles, the ratios of the corresponding sides are equal: $\frac{PQ}{MN} = \frac{QR}{NO}$

Substitute the known values: $\frac{x}{23} = \frac{56}{17}$

Now solve for $x$ (the length of side $PQ$): $x = \frac{56}{17} \times 23$

First, calculate $\frac{56}{17}$: $\frac{56}{17} \approx 3.2941$

Then, multiply by 23: $x = 3.2941 \times 23 \approx 75.7643$

Rounding to the nearest tenth, the length of side $PQ$ is approximately: $PQ \approx 75.8$

Would you like more details on this solution? Also, here are five related questions to explore this concept further:

1. What are the properties of similar triangles?
2. How do you find the ratio of corresponding sides in similar triangles?
3. Can the angles of similar triangles be used to determine unknown side lengths?
4. How would the solution change if another side of triangle PQR were given instead of $QR$?
5. What are some real-life applications of similar triangles in geometry?

Tip: In problems involving similar triangles, always make sure you correctly identify the corresponding sides before setting up the ratio!