Homework \#12 Distance on a Coordinate Grid \( \begin{array}{l}\text { 2. Find the distance } \\ \text { between the points } \\ (3,5) \text { and }(6,5)\end{array} \)

To find the distance between the two points \((3, 5)\) and \((6, 5)\) on a coordinate grid, you can use the **distance formula**. However, in this case, since both points share the same \(y\)-coordinate, the problem simplifies considerably. Here's a step-by-step explanation: ### Step 1: Identify the Coordinates - **Point A:** \((3, 5)\) - **Point B:** \((6, 5)\) ### Step 2: Understand the Position - Both points have the same \(y\)-coordinate (\(5\)), which means they lie on a **horizontal line**. ### Step 3: Use the Distance Formula The general distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the values: \[ \text{Distance} = \sqrt{(6 - 3)^2 + (5 - 5)^2} \] \[ \text{Distance} = \sqrt{(3)^2 + (0)^2} \] \[ \text{Distance} = \sqrt{9 + 0} \] \[ \text{Distance} = \sqrt{9} \] \[ \text{Distance} = 3 \] ### Step 4: Conclusion The distance between the points \((3, 5)\) and \((6, 5)\) is **3 units**. --- **Visualization:** Since both points are on the same horizontal line (\(y = 5\)), the distance is simply the difference in their \(x\)-coordinates. ``` Point A: (3,5) --------- Point B: (6,5) ``` The number of units between \(3\) and \(6\) on the \(x\)-axis is \(3\) units.