Question

find the coordinates of the orthocenter of △xyz with vertices x(3, -3), y(-1, -7), and z(-2, -3). the coordinates of the orthocenter are (□, □).

Explanation:

Step1: Find the slope of two sides

First, find the slope of \(XY\) and \(XZ\) (or other pairs). The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For \(X(3, -3)\) and \(Y(-1, -7)\): \(m_{XY}=\frac{-7 - (-3)}{-1 - 3}=\frac{-4}{-4} = 1\).
For \(X(3, -3)\) and \(Z(-2, -3)\): \(m_{XZ}=\frac{-3 - (-3)}{-2 - 3}=\frac{0}{-5}=0\) (horizontal line).

Step2: Find the slopes of the altitudes

The altitude to a side is perpendicular to that side. The slope of a perpendicular line is the negative reciprocal.

• Altitude to \(XZ\) (horizontal line) is vertical (undefined slope), so it's a vertical line through \(Y(-1, -7)\). The equation is \(x = -1\).
• Altitude to \(XY\): slope of \(XY\) is \(1\), so the slope of the altitude (perpendicular) is \(-1\). This altitude passes through \(Z(-2, -3)\). Using point - slope form \(y - y_1=m(x - x_1)\), we have \(y - (-3)=-1(x - (-2))\), which simplifies to \(y + 3=-x - 2\), or \(y=-x - 5\).

Step3: Find the intersection of the two altitudes

We have the equations \(x=-1\) and \(y=-x - 5\). Substitute \(x = -1\) into \(y=-x - 5\): \(y=-(-1)-5=1 - 5=-4\).

Answer:

\((-1, -4)\)