What are Parallel and Perpendicular Lines?

What are Parallel and Perpendicular Lines? exercise

• Describe the effect of the potions.

  Hints

  The general equation of a line is $y=mx+b$.

  • $m$ is the slope
  • $b$ the y-intercept

  Lines with the same slope are either equal or parallel.

  If the product of two slopes is equal to $-1$, the corresponding lines are perpendicular.

  Solution

  Richard the Mage is quite sad because his nemesis, the Dark Count, is taller than him.

  • Richard the Mage: 5 ft. tall
  • The Dark Count: 6 ft. tall

  Richard the Mage prepares a magic potion that will help him grow. The line that describes Richard's growth after he drinks the potion is $\frac14x+5$. The y-intercept is Richard's original height, 5 feet.

  The equation that describes the Dark Count's height is $y=6$.

  If the Dark Count drinks a magic growing potion as well, he will grow at the same rate as Richard the Mage. The equation describing the Dark Count's growth is $y=\frac14 x+6$. Since the slope of the lines describing Richard the Mage and the Dark Count's growths are now the same, we can say the lines are parallel.

  Richard prepares a magic shrinking potion for the Dark Count. The equation to describe the growth after he will take the shrinking potion is $y=-4x+40$. If the lines are perpendicular, the product of the slopeswill equal $-1$.

  $-4\times \frac14=-1$

  Perpendicular means that two lines meet to form a right angle.

• Determine the Dark Count's starting height without his Sunday hat.

  Hints

  Parallel lines have the same slope.

  Each point has an x- and y-coordinate: $P(p_x,p_y)$.

  The Distributive Property is:

  $a\times(b+c)=a\times b+a\times c$

  Solution

  The equation that describes the Dark Count's growth when he has on his Sunday hat is $y=\frac14x+b$. Remember, parallel lines have the same slope.

  We still have to examine the y-intercept. This is the Dark Count's starting height including the Sunday hat, which is unknown. We know his height on Day $12$, $12$ feet. So $x=12$ and $y=12$.

  We use the following point-slope-form:

  $y-y_1=m(x-x_1)$

  We substitute our known values, giving us:

  $y-12=\frac14(x-12)$

  Now we use the Distributive Property on the right side:

  $y-12=\frac14x-3$

  Adding $12$ on both sides give us the equation:

  $y=\frac14x+9$

  The Dark Count's starting height is our the y-intercept, 9 feet. So we also know the hat is 3 feet tall.

• Examine which line corresponding to the equation is parallel or perpendicular to the given line.

  Hints

  Two parallel lines have the same slope.

  Keep in mind that fractions can also be represented in decimal form.

  If the product of the slopes of two equations is equal to $-1$, then the corresponding lines are perpendicular.

  Solution

  Take a look at the equation $y=\frac14x+6=0.25x+6$.

  In order to decide if the equation of a line is parallel or perpendicular to a given line, you just have to look at the value for $m$.

  • If $m=\frac14=0.25$, the corresponding lines are parallel
  • If $m\times \frac14=-1$, the corresponding lines are perpendicular
  • If neither is true, then the corresponding lines are neither parallel nor perpendicular.

  1. The lines corresponding to $y=\frac14x+7$ and $y=0.25x+9$ are parallel to the line $y=\frac14x+6$.
  2. The lines to $y=-4x+22$ and $y=-4x+43$ are perpendicular to the line $y=\frac14x+6$.
  3. The other two equations and their corresponding lines are neither parallel nor perpendicular to the line $y=\frac14x+6$ because one slope is $0.5$ and the other slope is $\frac13$. Our original equation has a slope of $\frac14$.

• Determine the equation of a line that is perpendicular to the graph of the given equation.

  Hints

  Isolate $4$ in the equation $m\times \frac14=-1$ to find the slope of a line that would be perpendicular to the line with slope $m=\frac14$.

  Use the point-slope-form for linear equations, $y-y_1=m(x-x_1)$.

  The equation we're looking for is in the form $y=mx+b$ where $m$ is the slope and $b$ is the y-intercept.

  Solution

  We are looking for an equation $y=mx+b$ that is perpendicular to the line $y=\frac14 x+6$. What do we know?

  • The slope of a line that is perpendicular to $y=\frac14x+6$ will have a slope that is the negative inverse of $\frac1/4$.
  • The line passes through the point $P(7,14)$.

  First, we take a look at our slope $m$. Because the two lines are perpendicular, we know that $m\times \frac14=-1$.

  $\begin{array}{rcl} &~&\\ m\times\frac14&=&-1\\ \color{#669900}{\times 4} &&\color{#669900}{\times 4}\\ m&=&-4 \end{array}$

  Now we use the point-slope-form $y-y_1=m(x-x_1)$ and substitute the known slope and the x- and y-coordinates of point $P$.

  $y-14=-4(x-7)$

  Next, we use the Distributive Property on the right side of the equation:

  $y-14=-4x+28$

  Finally, we add $14$ to both sides of the equal sign to get the equation into slope-intercept form.

  $y=-4x+42$

• Decide if each line is parallel or perpendicular to the red line.

  Hints

  Parallel lines have the same slope. Therefore, they have the same steepness.

  The point where perpendicular lines meet forms a right angle.

  There is only one line that is parallel to the red line and one line that is perpendicular to the red line.

  Solution

  The red line belongs to the equation $y=\frac14x+5$.

  The green line is parallel to the red line. The corresponding equation is $y=\frac14x +6$.

  Notice that the slope of the two lines $m=\frac14$ is the same.

  The slope of the blue line is $m=-4$. The product of this slope and the slope of the red line, $\frac14$, is $-1$. So the blue line is perpendicular to the red line. The point where perpendicular lines meet forms a right angle.

  The orange and violet lines are neither parallel nor perpendicular to the red line.

• Write equations to describe the new streets.

  Hints

  If the slopes of two lines are the same, they are parallel. If the product of the two slopes is $-1$, the lines are perpendicular.

  Use the point-slope form for linear equations:

  $y-y_1=m(x-x_1)$

  $x_1$ and $y_1$ are the x- and y-coordinates of a given point.

  Use the Distributive Property

  $a\times (b+c)=a\times b+a\times c$

  The equation we're looking for is given by $y=mx+b$.

  • $m$ is the slope
  • $b$ is the y-intercept

  Solution

  We know the equation of one street is $y=2x+8$.

  We want to know the equation of a parallel street.

  • The slope is $m=2$ because the streets are parallel.
  • Now that we know the slope and one point, $P(6,6)$, we can substitute the known values into the equation $y-y_1=m(x-x_1)$.

  1. $x-6=2(x-6)$
  2. Use the Distributive Property to get: $y-6=2x-12$
  3. Use Opposite Operations and add $6$ to both sides: $y=2x-6$

  Now we look at the perpendicular street.

  • The slope is the negative reciprocal of $m=2$, which is $m=-0.5$. To check our work, we multiply the two slopes to see if they equal $-1$. $-0.5\times 2=-1$.
  • We're also given one point, $Q(1,10)$.
  • Now we substitute the known values into $y-y_1=m(x-x_1)$ just like we did in the previous problem.

  1. $y-10=-0.5(x-1)$
  2. Use the Distributive Property to get $y-10=-0.5x+0.5$
  3. Use Opposite Operations and add $10$ to both sides: $y=-0.5x+10.5$