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

Step 1: Note the two points.
The points are $(1,2)$ and $(4,6)$. We want the straight line distance between them.

Step 2: Recall the distance formula.
For points $(x_1,y_1)$ and $(x_2,y_2)$, the distance is \[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \] This comes from the Pythagorean theorem.

Step 3: Find the change in $x$.
The horizontal change is $x_2-x_1=4-1=3$.

Step 4: Find the change in $y$.
The vertical change is $y_2-y_1=6-2=4$.

Step 5: Square and add.
\[ d=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}. \]

Step 6: Take the root.
The square root of $25$ is $5$. This is the classic $3$-$4$-$5$ right triangle. \[ \boxed{5} \]