Question

The points P(2,0), Q(0,2), R(4,2) and S(2,4) are in the rectangular coordinate system. :
Column A: The distance from P to Q
Column B: The distance from R to S

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given

Hint:

Before calculating, you can sometimes visualize the points. The "run" (change in x) from P to Q is -2, and the "rise" (change in y) is 2. For R to S, the run is -2, and the rise is 2. Since the changes in x and y are the same in magnitude, the distances (hypotenuses of the right triangles) must be equal.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question requires the use of the distance formula to calculate the lengths of two line segments in a Cartesian coordinate system.
Step 2: Key Formula or Approach:
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Step 3: Detailed Explanation:
Calculate the distance for Column A (from P to Q):
P = (2, 0) and Q = (0, 2).
\[ d_{PQ} = \sqrt{(0 - 2)^2 + (2 - 0)^2} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \] Calculate the distance for Column B (from R to S):
R = (4, 2) and S = (2, 4).
\[ d_{RS} = \sqrt{(2 - 4)^2 + (4 - 2)^2} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \] Step 4: Comparing the Quantities:
Column A: \(\sqrt{8}\)
Column B: \(\sqrt{8}\)
The two distances are equal.