Chapter 11.2, Problem 4QR

To determine

To find:The value of $b$ so that segments $PQ$ and $RQ$ are perpendicular.

Answer to Problem 4QR

The required value of $b$ is $11$ .

Explanation of Solution

Given Information:

  $P=\left(1,2\right)$

  $Q=\left(5,3\right)$

Formula Used:

Slope of two perpendicular line is related as

  $m_1=-\frac{1}{m_2}$  ........ (1)

Here, $m_1,m_2$ are the slope of the two line.

Calculation:

Substitute $1,2$ for $x_1$ and $x_2$ , $5,3$ for $y_1$ and $y_2$ in equation (1)

  $\begin{array}{c}{m}_1=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{3-2}{5-1}\\ =\frac{1}{4}\end{array}$

If the two line are perpendicularthan

  $\begin{array}{c}{m}_1=-\frac{1}{m_2}\\ m_2=-\frac{1}{m_1}\\ m_2=-4\end{array}$

Therefore,

  $\begin{array}{c}-4=\frac{3-b}{5-3}\\ 3-b=-4\times 2\\ b=11\end{array}$

Conclusion:

Hence, the required value of $b$ is $11$