Question

Find an equation of the line passing through \((5,2)\) and the midpoint of the line segment joining \((-1,1)\) and \((3,9)\).

Short answer

The equation of the line passing through $(5,2)$ and the midpoint of the line segment joining $(-1,1)$ and $(3,9)$ is \(y = -\frac{3}{4}x + \frac{23}{4}\).

Step-by-step solution

Find the Midpoint

To find the midpoint of the line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\), use the midpoint formula: \((x',y') = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\) Our given points are: \((-1, 1)\) and \((3, 9)\). Apply the midpoint formula: \((x',y') = \left(\frac{-1 + 3}{2}, \frac{1 + 9}{2}\right) = (1, 5)\) The midpoint is \((1, 5)\).

Find the Slope

To find the slope of the line passing through the given point \((5, 2)\) and the midpoint \((1, 5)\), we'll use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Substitute the coordinates of the given points into the formula: \(m = \frac{5 - 2}{1 - 5} = \frac{3}{-4}\) The slope of the line is \(m=-\frac{3}{4}\).

Apply the Point-Slope Form

To find the equation of the line, we can use the point-slope form, which is: \(y - y_1 = m(x - x_1)\) To find the equation of the line passing through \((x_1, y_1) = (5, 2)\) with a slope \(m = -\frac{3}{4}\), we plug these values into the point-slope formula: \(y - 2 = -\frac{3}{4}(x - 5)\)

Simplify the Equation

Now, we need to simplify the equation to get it into the slope-intercept form (i.e., \(y = mx + b\)). First, distribute the slope to both terms in the parentheses: \(y - 2 = -\frac{3}{4}x + \frac{15}{4}\) Next, add 2 to both sides of the equation to isolate y: \(y = -\frac{3}{4}x + \frac{15}{4} + 2\) Combine the constant terms by converting 2 into a fraction with a denominator of 4: \(y = -\frac{3}{4}x + \frac{15}{4} + \frac{8}{4}\) Combine the fractions: \(y = -\frac{3}{4}x + \frac{23}{4}\) The equation of the line is \(y = -\frac{3}{4}x + \frac{23}{4}\).