To find the length of the interval $PQ$ between points $P$ and $Q$ on a coordinate plane, we can use the distance formula:

$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

From the graph:

• Point $P$ has coordinates $(3, 1)$.
• Point $Q$ has coordinates $(7, 5)$.

Now we can plug these coordinates into the formula:

$PQ = \sqrt{(7 - 3)^2 + (5 - 1)^2}$ $= \sqrt{4^2 + 4^2}$ $= \sqrt{16 + 16}$ $= \sqrt{32}$ $= \sqrt{16 \times 2} = 4\sqrt{2} \approx 5.7$

So, the length of $PQ$ is approximately $5.7$ (to 1 decimal place).

Would you like more details on this solution, or do you have any questions?

Here are five related questions to expand your understanding:

1. How do we derive the distance formula from the Pythagorean theorem?
2. What are some other applications of the distance formula in real-world scenarios?
3. How would the distance change if point $Q$ moved horizontally or vertically?
4. What would happen if the coordinates of $P$ or $Q$ were negative values?
5. Can you determine the midpoint of $PQ$ using its coordinates?

Tip: Always check the coordinates carefully before substituting values into the distance formula to avoid errors.