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The image shows a problem asking for the shortest distance between two points \( P(4,3) \) and \( Q(2,-1) \). The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of \( P(4,3) \) and \( Q(2,-1) \):

\[
d = \sqrt{(2 - 4)^2 + (-1 - 3)^2}
\]
\[
d = \sqrt{(-2)^2 + (-4)^2}
\]
\[
d = \sqrt{4 + 16}
\]
\[
d = \sqrt{20} = 2\sqrt{5}
\]

So, the shortest distance between points \( P \) and \( Q \) is \( 2\sqrt{5} \).

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How do you generalize the distance formula to 3D coordinates?
2. What is the midpoint between points \( P(4,3) \) and \( Q(2,-1) \)?
3. How do you calculate the slope of the line passing through points \( P \) and \( Q \)?
4. If point \( R \) lies on the line joining \( P \) and \( Q \), what would be the ratio of distances if \( R \) divides \( PQ \) internally?
5. How does the Pythagorean theorem relate to the distance formula in a coordinate plane?

**Tip**: When using the distance formula, always double-check the signs of the differences between coordinates to avoid mistakes.

The coordinates of points P and Q are (4,3) and (2,-1) respectively. Find the shortest distance between P and Q.

Find the shortest distance between points P(4,3) and Q(2,-1) using the distance formula in coordinate geometry. This problem applies the Pythagorean theorem to solve for the distance between two points on a plane, suitable for students in Grades 9-10.

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