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McGraw Hill Glencoe Geometry, 2012

MH

McGraw Hill Glencoe Geometry, 2012 View details

3. Slopes of Lines

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Exercise 32 Page 193

Calculate the slopes of the lines using the Slope Formula.

Are the Lines Parallel, Perpendicular, or Neither? Perpendicular
Graph:

Practice makes perfect

In order to determine the relationship between the lines, let's calculate each of their slopes.

Slope of $A B$

We are given the points

A (8, 4)

and

B (4, 3),

which lie on line

A B .

Let's substitute them into the Slope Formula and calculate the slope.

m_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(8, 4)$ & $(4, 3)$

m_{1} = \frac{3 - 4}{4 - 8}

Subtract terms

m_{1} = \frac{- 1}{- 4}

$\frac{- a}{- b} = \frac{a}{b}$

m_{1} = \frac{1}{4}

Slope of $C D$

Similarly, by substituting the points

C (4, - 9)

and

D (2, - 1)

into the Slope Formula, we can calculate the slope of the line

C D .

m_{2} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(4, - 9)$ & $(2, - 1)$

m_{2} = \frac{- 1 - ( - 9 )}{2 - 4}

$a - (- b) = a + b$

m_{2} = \frac{- 1 + 9}{2 - 4}

Add and subtract terms

m_{2} = \frac{8}{- 2}

MoveNegDenomToFrac

Put minus sign in front of fraction

m_{2} = - \frac{8}{2}

Calculate quotient

m_{2} = - 4

Analyze the Slopes

Now that we know the slopes of the lines, we can determine whether they are parallel, perpendicular, or neither. Remember that parallel slopes are the same and perpendicular slopes are opposite reciprocals.

m_{1} = \frac{1}{4} m_{2} = - 4

These slopes are opposite reciprocals, so the lines are perpendicular.

Graph

Let's graph the lines to verify that they are perpendicular. We can do this by plotting the points and drawing lines through them.

As we can see, the lines are truly perpendicular.

Subchapter links

Check Your Understanding p.192

Practice and Problem Solving p.193-195

H.O.T. Problems p.195

Standardized Test Practice p.196

Spiral Review p.196

Skills Review p.196