Chapter 14.3, Problem 16PSC

To determine

To find the orthocenter of the triangle.

Answer to Problem 16PSC

  $\left(4,5\right)$

Explanation of Solution

Given:

The triangle is RST with $R=(-3,2),S=(4,5)$ and $T=(7,-2)$

Calculation:

The orthocenter $O(x,y)$ of the triangle ABCis thepoint of concurrency of line containing altitudes.

Find an equation of a line that contains the altitude from the point T to the segment RS.

Slope of RS-

  $m=\frac{5-2}{4-(-3)}=\frac{3}{7}$

Find the slope of the altitude-

  $-\frac{1}{m}=-\frac{7}{3}$

Writing the equation of the altitude using point T-

  $\begin{array}{l}y-(-2)=-\frac{7}{3}(x-7)\\ y=-\frac{7}{3}x+\frac{43}{3}\\ \frac{3}{x}+\frac{23}{7}=-\frac{7}{3}x+\frac{43}{3}\\ 9x+69=-49x+301\\ 58x=232\\ x=4\\ y=\frac{3}{7}(4)+\frac{23}{7}=\frac{35}{7}=5\end{array}$

Therefore, the orthocenter is $\left(4,5\right)$