In the coordinate plane, two points A and B are plotted as shown in the figure.Point A has coordinates \((-1,...

1. TRANSLATE the problem information

From the graph and problem statement:

• Point A has coordinates \((-1, 2)\)
• Point B has coordinates \((5, -3)\)
• We need to find the distance between these two points

2. INFER the appropriate strategy

The distance between two points in the coordinate plane can be found using the distance formula, which comes from the Pythagorean theorem. Think of it this way: if you draw a right triangle with the line segment AB as the hypotenuse, the horizontal leg is the change in x-coordinates and the vertical leg is the change in y-coordinates.

Distance formula: \(\mathrm{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\)

3. SIMPLIFY to find the horizontal change

Calculate \(\Delta\mathrm{x}\) (change in x-coordinates):

• \(\Delta\mathrm{x} = \mathrm{x_2 - x_1} = 5 - (-1)\)
• \(\Delta\mathrm{x} = 5 + 1 = 6\)

Watch out! When subtracting a negative number, it becomes addition: \(5 - (-1) = 5 + 1\)

4. SIMPLIFY to find the vertical change

Calculate \(\Delta\mathrm{y}\) (change in y-coordinates):

• \(\Delta\mathrm{y} = \mathrm{y_2 - y_1} = -3 - 2\)
• \(\Delta\mathrm{y} = -5\)

5. SIMPLIFY using the distance formula

Now substitute into the formula:

• \(\mathrm{d} = \sqrt{(6)^2 + (-5)^2}\)
• \(\mathrm{d} = \sqrt{36 + 25}\)
• \(\mathrm{d} = \sqrt{61}\)

Note: \((-5)^2 = 25\) (positive), not -25

6. SIMPLIFY to approximate the square root

We need to find \(\sqrt{61}\) (use calculator):

• \(\sqrt{61} \approx 7.81\)

Alternatively, estimate: Since \(\sqrt{64} = 8\) and 61 is slightly less than 64, \(\sqrt{61}\) should be slightly less than 8.

7. APPLY CONSTRAINTS to select the closest answer

Compare 7.81 to the answer choices:

• (A) 6.2 — too far below
• (B) 7.1 — about 0.7 away
• (C) 7.8 — only 0.01 away ✓
• (D) 9.2 — way too high

Answer: C (7.8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors with negative numbers

When calculating \(\Delta\mathrm{x} = 5 - (-1)\), students may incorrectly compute this as:

• \(5 - 1 = 4\) (forgetting that subtracting a negative becomes addition)

Or when calculating \(\Delta\mathrm{y} = -3 - 2\), they might get:

• \(-3 - 2 = -1\) (incorrect arithmetic with negatives)

If a student uses \(\Delta\mathrm{x} = 4\) and \(\Delta\mathrm{y} = -5\):

• \(\mathrm{Distance} = \sqrt{(4)^2 + (-5)^2}\)
• \(= \sqrt{16 + 25}\)
• \(= \sqrt{41} \approx 6.4\)

This may lead them to select Choice A (6.2) as the closest value.

Second Most Common Error:

Weak SIMPLIFY skill: Squaring negative numbers incorrectly

When squaring \((-5)\), students might incorrectly calculate:

• \((-5)^2 = -25\) (forgetting that negative times negative equals positive)

This creates confusion because you cannot take the square root of a negative number in this context, leading them to make additional errors or guess. Even if they proceed with \(\sqrt{36 - 25} = \sqrt{11} \approx 3.3\), this doesn't match any answer choice well, causing confusion and potentially leading to a random selection like Choice B (7.1).

The Bottom Line:

This problem tests whether students can carefully execute multi-step calculations involving negative numbers. The conceptual part (knowing the distance formula) is straightforward, but the execution requires attention to detail with sign conventions. Most errors stem from rushing through the arithmetic rather than from not knowing the formula.