Question
Question 1: The midpoint of \( (a, 4) \) and \( (6, b) \) is \( (5,8) \). The values of \( a \) and \( b \) are respectively
Ask by Reeves Jimenez. in Australia
Jan 17,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( a = 4 \) and \( b = 12 \).
Solution
To find the values of \( a \) and \( b \), we'll use the **midpoint formula**. The midpoint \( (M_x, M_y) \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2}
\]
Given:
- The two points are \( (a, 4) \) and \( (6, b) \).
- The midpoint is \( (5, 8) \).
**Step 1: Find \( a \)**
Using the formula for the x-coordinate of the midpoint:
\[
\frac{a + 6}{2} = 5
\]
Multiply both sides by 2:
\[
a + 6 = 10
\]
Subtract 6 from both sides:
\[
a = 4
\]
**Step 2: Find \( b \)**
Using the formula for the y-coordinate of the midpoint:
\[
\frac{4 + b}{2} = 8
\]
Multiply both sides by 2:
\[
4 + b = 16
\]
Subtract 4 from both sides:
\[
b = 12
\]
**Conclusion:**
The values are \( a = 4 \) and \( b = 12 \).
**Answer:**
\( a = 4 \) and \( b = 12 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find the values of \( a \) and \( b \), we can use the formula for the midpoint. The midpoint \( (M_x, M_y) \) of points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as: \[ M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2} \] Given the points \( (a, 4) \) and \( (6, b) \), the midpoint is given as \( (5, 8) \). From \( M_x = 5 \): \[ \frac{a + 6}{2} = 5 \] Multiplying both sides by 2: \[ a + 6 = 10 \] Subtracting 6 from both sides gives: \[ a = 4 \] From \( M_y = 8 \): \[ \frac{4 + b}{2} = 8 \] Multiplying both sides by 2: \[ 4 + b = 16 \] Subtracting 4 from both sides gives: \[ b = 12 \] Thus, the values of \( a \) and \( b \) are respectively \( 4 \) and \( 12 \).
