Question

The distance between the points \[ (1,2) \quad \text{and} \quad (4,6) \] is:

• A. \(3\)
• B. \(4\)
• C. \(5\)
• D. \(6\)

Hint:

Remember the famous Pythagorean triple \[ 3,\;4,\;5. \] Whenever coordinate differences are \(3\) and \(4\), the distance is immediately \(5\).

Answer 1

The Correct Option is C

Concept: The distance formula is one of the most important formulas in coordinate geometry. It is derived from the Pythagorean theorem and gives the shortest distance between two points in a plane. For points \[ (x_1,y_1) \] and \[ (x_2,y_2), \] the distance between them is \[ d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \] This formula converts a geometric problem into a simple algebraic computation.

Step 1: Identify the coordinates. The given points are \[ (1,2) \] and \[ (4,6). \] Therefore, \[ x_1=1,\quad y_1=2, \] \[ x_2=4,\quad y_2=6. \]

Step 2: Apply the distance formula. \[ d= \sqrt{(4-1)^2+(6-2)^2}. \]

Step 3: Simplify the coordinate differences. \[ = \sqrt{3^2+4^2}. \] \[ = \sqrt{9+16}. \]

Step 4: Add the squares. \[ = \sqrt{25}. \] \[ =5. \]

Step 5: Verification. The horizontal difference is \(3\) units and the vertical difference is \(4\) units. These form a right triangle whose hypotenuse is \[ \sqrt{3^2+4^2}=5. \] Thus the answer is verified.

Step 6: Final Conclusion. \[ \boxed{5} \] Hence the correct answer is \[ \boxed{\text{Option (C)}}. \]