Animated Solution for Mathematics - Straight Lines: If a $\Delta ABC$ has vertices $A(-1,7)$, $B(-7,1)$ and $C(5,-5)$, then its orthocentre has coordinates

Finding the Orthocentre

Welcome, future engineer! Today, we are not just solving a coordinate geometry problem; we are uncovering the hidden architecture of a triangle.

We are tasked with finding the orthocentre of $\Delta ABC$ with vertices $A(-1,7)$, $B(-7,1)$, and $C(5,-5)$. The orthocentre, denoted as $H$, is the point where the three altitudes of a triangle meet. It is a point of perfect balance, a geometric center of gravity for the altitudes.

The Logic of Perpendicularity

To find the intersection of two lines, we need their equations. An altitude is a line segment from a vertex perpendicular to the opposite side.

Let's start with the altitude from vertex $A$ to side $BC$. To find its equation, we first need the slope of side $BC$. Using the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$, we substitute the coordinates of $B(-7,1)$ and $C(5,-5)$:

Now, here is the magic. Since our altitude $AD$ is perpendicular to $BC$, their slopes must satisfy the condition $m_{AD}\cdot m_{BC}=-1$.

Thus, the slope of $AD$ is the negative reciprocal of $-\frac{1}{2}$, which is $2$. With a point $A(-1,7)$ and a slope $m=2$, we use the point-slope form $y-y_1=m(x-x_1)$:

The Second Altitude

We repeat this elegant process for the altitude from vertex $B$ to side $AC$. First, we calculate the slope of $AC$:

Since the altitude $BE$ is perpendicular to $AC$, its slope $m_{BE}$ must be the negative reciprocal of $-2$, which is $\frac{1}{2}$. Now, using point $B(-7,1)$ and slope $\frac{1}{2}$:

The Intersection

We now have a system of two linear equations:

1) $2x-y+9=0$

2) $x-2y+9=0$

From the first equation, we can write $y=2x+9$. Substituting this into the second equation:

Finally, substituting $x=-3$ back into $y=2x+9$, we get $y=2(-3)+9=3$.

The orthocentre $H$ is at $(-3,3)$. You have successfully navigated the geometry and the algebra to find the heart of the triangle. Keep this systematic approach in your toolkit, and no coordinate geometry problem will ever stand in your way!