12 multiple-choice questions, progressively harder.
A line has slope 444. What is the slope of any line parallel to it?
Solution
Correct answer: C
Parallel lines point in the same direction, so they have equal slopes.
m∥=4m_{\parallel} = 4m∥=4
Any line parallel to a line of slope 444 also has slope 444.
Two different lines have the same slope. How are they related?
Correct answer: D
Equal slopes mean equal steepness, so the two lines never draw closer or farther apart.
m1=m2 ⇒ parallelm_1 = m_2 \;\Rightarrow\; \text{parallel}m1=m2⇒parallel
Since the lines are stated to be different (different intercepts), they run alongside each other without meeting, so they are parallel.
A line has slope −2-2−2. What is the slope of any line parallel to it?
Correct answer: B
Parallel lines share the same slope, sign and all.
m∥=−2m_{\parallel} = -2m∥=−2
So a line parallel to a line of slope −2-2−2 also has slope −2-2−2.
What is the slope of a line perpendicular to a line with slope 12\tfrac{1}{2}21?
Correct answer: A
Flip 12\tfrac{1}{2}21 to 222, then change the sign to get the negative reciprocal.
m⊥=−2m_{\perp} = -2m⊥=−2
As a check, 12⋅(−2)=−1\tfrac{1}{2} \cdot (-2) = -121⋅(−2)=−1.
A line has slope −1-1−1. What is the slope of a line perpendicular to it?
The reciprocal of −1-1−1 is −1-1−1; changing the sign gives 111.
m⊥=−1−1=1m_{\perp} = -\frac{1}{-1} = 1m⊥=−−11=1
And (−1)(1)=−1(-1)(1) = -1(−1)(1)=−1, so a line of slope 111 is perpendicular to a line of slope −1-1−1.
Two different lines both have slope 666. Are they parallel, perpendicular, or neither?
The slopes are equal, and the lines are stated to be different.
m1=m2=6 ⇒ parallelm_1 = m_2 = 6 \;\Rightarrow\; \text{parallel}m1=m2=6⇒parallel
Equal slopes with different intercepts means the lines never meet, so they are parallel.
What is the negative reciprocal of 444?
The reciprocal of 4=414 = \tfrac{4}{1}4=14 is 14\tfrac{1}{4}41; the negative reciprocal changes the sign.
−14-\frac{1}{4}−41
This is the slope perpendicular to a line of slope 444.
A line has slope −15-\tfrac{1}{5}−51. What is the slope of a line perpendicular to it?
Flip −15-\tfrac{1}{5}−51 to −5-5−5, then change the sign for the negative reciprocal.
m⊥=5m_{\perp} = 5m⊥=5
As a check, −15⋅5=−1-\tfrac{1}{5} \cdot 5 = -1−51⋅5=−1.
Perpendicular lines meet at what kind of angle?
Perpendicular is the name for lines that cross squarely, like the corner of a page.
perpendicular ⇒ 90∘ angle\text{perpendicular} \;\Rightarrow\; 90^{\circ} \text{ angle}perpendicular⇒90∘ angle
So perpendicular lines meet at a right angle of 909090 degrees.
A horizontal line and a vertical line are which of the following?
Think of the xxx-axis and the yyy-axis: one runs flat, the other stands straight up, and they cross at a right angle.
horizontal⊥vertical\text{horizontal} \perp \text{vertical}horizontal⊥vertical
So a horizontal line and a vertical line are perpendicular. This pair is checked by sight, since a vertical line has no slope to put in the product rule.
What is the negative reciprocal of −13-\tfrac{1}{3}−31?
The reciprocal of −13-\tfrac{1}{3}−31 is −3-3−3; changing the sign gives the negative reciprocal.
−1−13=3-\frac{1}{-\tfrac{1}{3}} = 3−−311=3
So the negative reciprocal of −13-\tfrac{1}{3}−31 is 333.
The lines y=−x+2y = -x + 2y=−x+2 and y=x+2y = x + 2y=x+2 are which of the following?
Their slopes are −1-1−1 and 111. Multiply them and compare with −1-1−1.
(−1)(1)=−1(-1)(1) = -1(−1)(1)=−1
The product is −1-1−1, so the lines are perpendicular. They happen to cross at the shared intercept (0,2)(0, 2)(0,2), and they cross there at a right angle.
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