Question

The distance between the points $ (0, 0) $ and $ (6, 5) $ is

• A. $ \sqrt{22} $
• B. $ \sqrt{61} $
• C. $ \sqrt{11} $
• D. 9

Hint:

When one point is the origin, directly use $d = \sqrt{x^2 + y^2}$ to save time in calculations.

Answer 1

The Correct Option is B

Concept: The distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ in a Cartesian plane is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula is derived from the Pythagorean theorem by treating the horizontal and vertical differences as perpendicular sides of a right triangle.

Step 1: Identify the coordinates.
\[ (x_1,y_1) = (0,0), \quad (x_2,y_2) = (6,5) \]

Step 2: Apply the distance formula.
\[ d = \sqrt{(6-0)^2 + (5-0)^2} \]

Step 3: Simplify each term.
\[ d = \sqrt{6^2 + 5^2} \] \[ d = \sqrt{36 + 25} \]

Step 4: Final computation.
\[ d = \sqrt{61} \] Since 61 has no perfect square factors, the result cannot be simplified further. Conclusion: \[ \boxed{\sqrt{61}} \]