Chapter 13.9, Problem 5CE

(a)

To determine

To find: The equation of line k .

Answer to Problem 5CE

The equation of line k is $x=b$ .

Explanation of Solution

Given information: The lines j , k and l contains the altitudes of $\Delta ROM$ as shown below.

  

Calculation:

Let the line k cut the x- axis at the point P . From the figure it can be observed that the point P has coordinates $\left(b,0\right)$ .

It can be seen that the line k is a vertical line or parallel to y -axis. It passes through the points $\left(b,c\right)$ and $\left(b,0\right)$ . So, the equation of line k is:

  $x=b$

Therefore, the equation of line k is $x=b$ .

(b)

To determine

To find: The slope of the line l .

Answer to Problem 5CE

The slope of the line l is $\frac{a-b}{c}$ .

Explanation of Solution

Given information: The lines j , k and l contains the altitudes of $\Delta ROM$ as shown below.

  

The slope of the line $\overline{MR}$ is $\frac{c}{b-a}$ .

Calculation:

It is given that the line l contains the altitude of $\Delta ROM$ . Altitude is perpendicular to the line $\overline{MR}$ .

The slope of line $\overline{MR}$ is $m_1=\frac{c}{b-a}$ .

The condition for two perpendicular lines is given by,

  $m_1{m}_2=-1$

The slope of the line l can be calculated as:

  $\begin{array}{c}\left(\frac{c}{b-a}\right)m_2=-1\\ m_2=\frac{a-b}{c}\end{array}$

Therefore, the slope of the line l is $\frac{a-b}{c}$ .

(c)

To determine

To show: The equation of the line l is $y=\left(\frac{a-b}{c}\right)x$ .

Explanation of Solution

Given information: The lines j , k and l contains the altitudes of $\Delta ROM$ as shown below.

  

Proof:

From part (b), slope of the line l is $\frac{a-b}{c}$ . From the figure it can be seen that the line l passes through the origin.

The formula to find the equation of a line with given slope that passes through a point is given by,

  $y-y_1=m\left(x-x_1\right)$

Substitute 0 for $x_1$ , 0 for $y_1$ and $\frac{a-b}{c}$ for $m$ in the above formula.

  $\begin{array}{c}y-0=\frac{a-b}{c}\left(x-0\right)\\ y=\frac{a-b}{c}x\end{array}$

Hence, it is proved that the equation of the line l is $y=\left(\frac{a-b}{c}\right)x$ .

(d)

To determine

To show: The lines k and l intersect at $x=b$ and $y=\frac{ab-b^2}{c}$ .

Explanation of Solution

Given information: The lines j , k and l contains the altitudes of $\Delta ROM$ as shown below.

  

Proof:

From part (a), the equation of line k is $x=b$ . From part (d), the equation of the line l is $y=\left(\frac{a-b}{c}\right)x$ .

To find y -coordinate, substitute b for x in the equation of line l .

  $\begin{array}{c}y=\left(\frac{a-b}{c}\right)b\\ y=\frac{ab-b^2}{c}\end{array}$

Hence, it is proved that the lines k and l intersect at $x=b$ and $y=\frac{ab-b^2}{c}$ .

(e)

To determine

To find: The slope of the line $\overline{OM}$ and the slope of line $j$ .

Answer to Problem 5CE

The slope of the line $\overline{OM}$ is $\frac{c}{b}$ and the slope of line $j$ is $\frac{-b}{c}$ .

Explanation of Solution

Given information: The lines j , k and l contains the altitudes of $\Delta ROM$ as shown below.

  

Calculation:

The slope of the line $\overline{OM}$ that passes through the points $\left(0,0\right)$ and $\left(b,c\right)$ is:

  $\begin{array}{c}{m}_1=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{c-0}{b-0}\\ =\frac{c}{b}\end{array}$

The condition for two perpendicular lines is given by,

  $m_1{m}_2=-1$

From the figure it can be observed that line j is perpendicular to $\overline{OM}$ . So, the slope of the line j can be calculated as:

  $\begin{array}{c}\left(\frac{c}{b}\right)m_2=-1\\ m_2=\frac{-b}{c}\end{array}$

Therefore, the slope of the line $\overline{OM}$ is $\frac{c}{b}$ and the slope of line $j$ is $\frac{-b}{c}$ .

(f)

To determine

To show: The equation of the line j is $y=\frac{-b}{c}\left(x-a\right)$ .

Explanation of Solution

Given information: The lines j , k and l contains the altitudes of $\Delta ROM$ as shown below.

  

Proof:

From part (e), slope of the line j is $\frac{-b}{c}$ . From the figure it can be seen that the line j passes through the point $\left(a,0\right)$ .

The formula to find the equation of a line with given slope that passes through a point is given by,

  $y-y_1=m\left(x-x_1\right)$

Substitute a for $x_1$ , 0 for $y_1$ and $\frac{-b}{c}$ for $m$ in the above formula.

  $\begin{array}{c}y-0=\frac{-b}{c}\left(x-a\right)\\ y=\frac{-b}{c}\left(x-a\right)\end{array}$

Hence, it is proved that the equation of the line j is $y=\frac{-b}{c}\left(x-a\right)$ .

(g)

To determine

To show: The lines k and j intersect at $x=b$ and $y=\frac{ab-b^2}{c}$ .

Explanation of Solution

Given information: The lines j , k and l contains the altitudes of $\Delta ROM$ as shown below.

  

Proof:

From part (a), the equation of line k is $x=b$ . From part (d), the equation of the line j is $y=\frac{-b}{c}\left(x-a\right)$ .

To find y -coordinate, substitute b for x in the equation of line j .

  $\begin{array}{c}y=\frac{-b}{c}\left(b-a\right)\\ y=\frac{ab-b^2}{c}\end{array}$

Hence, it is proved that the lines k and j intersect at $x=b$ and $y=\frac{ab-b^2}{c}$ .

(h)

To determine

To write: The coordinates of three altitude lines.

Explanation of Solution

From part (d) and (g), the lines k , l and j intersect at $x=b$ and $y=\frac{ab-b^2}{c}$ .

Therefore, the coordinates of the point where three altitude lines intersect is $\left(b,\frac{ab-b^2}{c}\right)$ .