Animated Solution for Mathematics - Straight Lines: Let $PS$ be the median of the triangle with vertices $P(2,2),Q(6,-1)$ and $R(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to $PS$ is

Have you ever looked at a triangle on a coordinate plane and felt like it was just a collection of static points? Today, we are going to change that. We are going to breathe life into the triangle $PQR$ with vertices $P(2,2)$, $Q(6,-1)$, and $R(7,3)$.

Our mission is to find the equation of a line that passes through $(1,-1)$ and dances in perfect harmony—parallel—to the median $PS$.

A median is not just any line; it is a line of symmetry, a bridge from a vertex to the exact center of the opposite side. To find the median $PS$, we first need to locate point $S$, the midpoint of $QR$.

Think of the midpoint formula as the average of the coordinates. We take the $x$-coordinates of $Q$ and $R$, which are $6$ and $7$, and find their average:

Then, we do the same for the $y$-coordinates, $-1$ and $3$:

Just like that, we have found the heart of the side $QR$: $S(\frac{13}{2},1)$.

Now that we have $P(2,2)$ and $S(\frac{13}{2},1)$, we can define the slope of our median $PS$. The slope is the measure of how steep our line is, the ratio of the vertical rise to the horizontal run.

Using the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$, we calculate:

The numerator is $-1$. The denominator, $\frac{13}{2}-2$, simplifies to $\frac{13-4}{2}=\frac{9}{2}$.

Dividing $-1$ by $\frac{9}{2}$ gives us the slope:

This is the DNA of our line—its direction.

Here is where the magic happens. We are told our target line is parallel to $PS$. In the world of coordinate geometry, parallel lines are twins; they share the exact same slope.

So, our target line also has a slope of $m=-\frac{2}{9}$. We know it passes through $(1,-1)$.

We have a point and a slope—this is the perfect setup for the point-slope form: $y-y_1=m(x-x_1)$. Substituting our values, we get:

Now, let's bring this to its standard form. First, simplify the left side: $y+1=-\frac{2}{9}(x-1)$.

To clear the fraction, multiply both sides by $9$:

Expanding this, we get $9y+9=-2x+2$. Finally, move everything to one side to reach the standard form:

This simplifies beautifully to the final equation:

You have successfully navigated the geometry, calculated the midpoint, determined the slope, and constructed the equation. You didn't just solve a problem; you mapped a path through the coordinate plane.