Question
Calculate the distance between the points \( (3, 4) \) and \( (7, 1) \).
Ask by Tran Mann. in India
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The distance between the points \( (3, 4) \) and \( (7, 1) \) is 5 units.
Solution
To calculate the distance between two points in a Cartesian plane, you can use the **Distance Formula**, which is derived from the Pythagorean theorem.
Given two points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \), the distance \( d \) between them is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's apply this formula to your specific points \( (3, 4) \) and \( (7, 1) \).
1. **Identify the coordinates:**
- \( P_1 = (3, 4) \) where \( x_1 = 3 \) and \( y_1 = 4 \)
- \( P_2 = (7, 1) \) where \( x_2 = 7 \) and \( y_2 = 1 \)
2. **Subtract the corresponding coordinates:**
- \( x_2 - x_1 = 7 - 3 = 4 \)
- \( y_2 - y_1 = 1 - 4 = -3 \)
3. **Square these differences:**
- \( (x_2 - x_1)^2 = 4^2 = 16 \)
- \( (y_2 - y_1)^2 = (-3)^2 = 9 \)
4. **Add the squared differences:**
- \( 16 + 9 = 25 \)
5. **Take the square root of the sum to find the distance:**
- \( d = \sqrt{25} = 5 \)
**Therefore, the distance between the points \( (3, 4) \) and \( (7, 1) \) is 5 units.**
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the distance between the points \( (3, 4) \) and \( (7, 1) \), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the values: - \( x_1 = 3 \), \( y_1 = 4 \) - \( x_2 = 7 \), \( y_2 = 1 \) The distance calculation would look like this: \[ d = \sqrt{(7 - 3)^2 + (1 - 4)^2} \] \[ d = \sqrt{(4)^2 + (-3)^2} \] \[ d = \sqrt{16 + 9} \] \[ d = \sqrt{25} \] \[ d = 5 \] So, the distance between the points \( (3, 4) \) and \( (7, 1) \) is \( 5 \) units.
