Chapter 8.4, Problem 43WE

To determine

To Find: The value of $t$ .

Answer to Problem 43WE

The value of t is 3.

Explanation of Solution

Given information:

Line joining C to $\left(-3t,2t+1\right)$ is parallel to line joining B and C.

Where $A\left(5,3\right),B\left(2,6\right),C\left(-2,0\right)$ .

Formula used: -

Slope of line passing through $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$

  $m=\frac{y_2-y_1}{x_2-x_1}$

Slope of parallel lines with slope $m_1$ and $m_2$ is $m_1=m_2$

Slope of line joining $C\left(-2,0\right)$ to $\left(-3t,2t+1\right)$

  $\begin{array}{l}{m}_1=\frac{y_2-y_1}{x_2-x_1}\\ m_1=\frac{2t+1-0}{-3t-\left(-2\right)}\\ m_1=\frac{2t+1}{-3t+2}\end{array}$

Slope of line joining $A\left(5,3\right),B\left(2,6\right)$ .

  $\begin{array}{l}m=\frac{6-3}{2-5}\\ m=\frac{3}{-3}\\ m=-1\end{array}$

As we know, Slope of parallel lines with slope $m_1$ and $m_2$ is $m_1=m_2$ .

Line joining A to $\left(-3t,2t+1\right)$ is parallel to line joining B and A.

  $\begin{array}{l}{m}_1=m_2\\ \frac{2t+1}{-3t+2}=-1\\ 2t+1=-1\left(-3t+2\right)\\ 2t+1=3t-2\end{array}$

  $\begin{array}{l}1+2=3t-2t\\ t=3\end{array}$

The value of t is 3.