What is the distance between the points (3, 6) and (-2, -6) in the Cartesian plane?

Introduction / Context:
This question tests knowledge of the distance formula in coordinate geometry. You are asked to find the straight-line distance between two points in the Cartesian plane, namely (3, 6) and (-2, -6). Such questions appear frequently in aptitude tests, engineering entrance exams and school mathematics, and they build intuition for how distances are measured using coordinates.

Given Data / Assumptions:

• First point: P(3, 6).
• Second point: Q(-2, -6).
• We are asked for the distance between P and Q.
• The distance is to be interpreted as the Euclidean distance (straight-line distance).

Concept / Approach:
The standard distance formula between two points (x1, y1) and (x2, y2) in the Cartesian plane is:
distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2] This formula comes directly from the Pythagorean theorem, considering the line segment joining the two points as the hypotenuse of a right triangle whose legs represent the horizontal and vertical differences in coordinates.

Step-by-Step Solution:
Step 1: Identify x1, y1, x2 and y2. Let P(3, 6) be (x1, y1) and Q(-2, -6) be (x2, y2). So x1 = 3, y1 = 6, x2 = -2, y2 = -6. Step 2: Compute the differences in x and y. x2 - x1 = -2 - 3 = -5. y2 - y1 = -6 - 6 = -12. Step 3: Square the differences. (x2 - x1)^2 = (-5)^2 = 25. (y2 - y1)^2 = (-12)^2 = 144. Step 4: Add and take the square root. distance = sqrt[25 + 144] = sqrt[169]. sqrt[169] = 13. Therefore, the distance between the two points is 13 units.
Verification / Alternative check:
We can interpret the movement from P(3, 6) to Q(-2, -6) as moving 5 units left (from x = 3 to x = -2) and 12 units down (from y = 6 to y = -6). This forms a right triangle with legs 5 and 12. A well-known Pythagorean triple is (5, 12, 13), meaning a right triangle with legs 5 and 12 has a hypotenuse of 13. This matches our calculated distance, confirming the result.

Why Other Options Are Wrong:
11 units and 12 units are both less than the true distance and do not satisfy the Pythagorean relation 5^2 + 12^2 = 169.
10 units would correspond to a much smaller hypotenuse and is clearly inconsistent with legs of 5 and 12.
15 units is larger than required and would correspond to a different set of leg lengths.
Only 13 units exactly satisfies the distance formula for the given coordinates.

Common Pitfalls:
Students sometimes forget to square the differences, or they mistakenly add the coordinates instead of subtracting them. Another error is to take the absolute values of differences and then add without squaring, which is incorrect. Additionally, some learners forget to take the square root after summing the squares, leaving the squared distance instead of the actual distance. Carefully following the distance formula step by step avoids all of these mistakes.

Final Answer:
The distance between the points (3, 6) and (-2, -6) is 13 units.