Find the equation of the line which passes through the point (−2, 2) and the midpoint of the line segment joining the points (4, 3) and (−2, 5)

To find the equation of the line passing through the point \((-2, 2)\) and the midpoint of the line segment joining the points \((4, 3)\) and \((-2, 5)\), we can follow these steps:

### Step 1: Find the Midpoint of the Line Segment

The midpoint \((M)\) of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

Substituting the given points \((4, 3)\) and \((-2, 5)\):

\[
M = \left(\frac{4 + (-2)}{2}, \frac{3 + 5}{2}\right) = \left(\frac{2}{2}, \frac{8}{2}\right) = (1, 4)
\]

### Step 2: Find the Slope of the Line

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((-2, 2)\) and \((1, 4)\):

\[
m = \frac{4 - 2}{1 - (-2)} = \frac{2}{3}
\]

### Step 3: Use the Point-Slope Form to Find the Equation

The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

Using the slope \(m = \frac{2}{3}\) and the point \((-2, 2)\):

\[
y - 2 = \frac{2}{3}(x - (-2))
\]

Simplify this equation:

\[
y - 2 = \frac{2}{3}(x + 2)
\]

\[
y - 2 = \frac{2}{3}x + \frac{4}{3}
\]

\[
y = \frac{2}{3}x + \frac{4}{3} + 2
\]

\[
y = \frac{2}{3}x + \frac{4}{3} + \frac{6}{3}
\]

\[
y = \frac{2}{3}x + \frac{10}{3}
\]

### Final Equation of the Line

\[
\boxed{y = \frac{2}{3}x + \frac{10}{3}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How to find the equation of a line given two points directly?
2. What is the significance of the midpoint in coordinate geometry?
3. How to convert a point-slope form to slope-intercept form?
4. How does changing one point affect the slope of the line?
5. What are other forms of linear equations besides slope-intercept and point-slope?

**Tip:** Always double-check the calculated midpoint and slope before forming the final equation to avoid simple calculation errors!

Find the equation of the line which passes through the point (−2, 2) and the midpoint of the line segment joining the points (4, 3) and (−2, 5).

Learn how to find the equation of a line passing through the point (-2, 2) and the midpoint of a line segment joining the points (4, 3) and (-2, 5). This problem covers key concepts in coordinate geometry, including the midpoint formula, slope of a line, and point-slope form. Suitable for students in Grades 8-10.

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