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| 14 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Two coplanar lines — lines that are on the same plane — that do not intersect are said to be parallel lines. In a diagram, triangular hatch marks are drawn on lines to denote that they are parallel. The symbol ∥
is used to algebraically denote that two lines are parallel. In the diagram, lines m and ℓ are parallel.
In a coordinate plane, two distinct non-vertical lines are parallel if and only if their slopes are equal.
If ℓ1 and ℓ2 are two parallel lines and m1 and m2 their respective slopes, then the following statement is true.
ℓ1∥ℓ2⇔m1=m2
The slope of a vertical line is not defined. Therefore, this theorem only applies to non-vertical lines. However, any two distinct vertical lines are parallel.
Since the theorem consists of a biconditional statement, the proof consists of two parts.
(I): y=m2x+b2
(II): x=m2−m1b1−b2
ℓ1∥ℓ2⇒m1=m2
(I): y=mx+b2
(I): LHS−mx=RHS−mx
m1=m2⇒ℓ1∥ℓ2
Both directions of the biconditional statement have been proved.
ℓ1∥ℓ2⇔m1=m2
Consider two lines on the same coordinate plane and their equations in slope-intercept form. Are the lines parallel?
Write an equation in slope-intercept form of the line that passes through (-2,3) and is parallel to y=2x−1.
What do parallel lines have in common?
As seen above, the graph of y=2x+7 passes through (-2,3) and is parallel to the graph of y=2x−1.
Find the slope of the line shown on the graph.
Substitute (0,3) & (1,1)
Subtract terms
1a=a
As seen above, the graph of y=-2x−4.5 is parallel to the pavement and passes through (-4,3.5).
Zosia wants to propose a new mural to be painted on the side of the planetarium. She starts with a moon and two stars that are already painted on the building.
Zosia wants to place more stars in the line that connects the two existing stars. She also wants to make a second line of stars that is parallel to the first and passes through the moon. The two stars and the moon can be represented on a coordinate plane.
Find the slope of the line that passes through the stars.
Substitute (-4,-3) & (3,0.5)
a−(-b)=a+b
Add terms
ba=b⋅2a⋅2
ba=b/7a/7
x=-2, y=1
b1⋅a=ba
Put minus sign in front of fraction
aa=1
LHS+1=RHS+1
Rearrange equation
Two coplanar lines — lines that are on the same plane — that intersect at a right angle are said to be perpendicular lines. The symbol ⊥
is used to algebraically denote that two lines are perpendicular. In the diagram, lines m and ℓ are perpendicular.
In a coordinate plane, two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals.
If ℓ1 and ℓ2 are two perpendicular lines and m1 and m2 their respective slopes, the following relation holds true.
ℓ1⊥ℓ2⇔m1⋅m2=-1
This theorem does not apply to vertical lines because their slope is undefined. However, vertical lines are always perpendicular to horizontal lines.
Since the theorem is a biconditional statement, the proof consists of two parts.
Side | Points | Distance Formula(x2−x1)2+(y2−y1)2
|
Length |
---|---|---|---|
AO | A(1,m1) & O(0,0) | (0−1)2+(0−m1)2 | 1+m12 |
CO | C(1,m2) & O(0,0) | (0−0)2+(0−m2)2 | 1+m22 |
CA | C(1,m2) & A(1,m1) | (1−1)2+(m1−m2)2 | m1−m2 |
Substitute expressions
(a)2=a
Add terms
(a−b)2=a2−2ab+b2
LHS−m12=RHS−m12
LHS−m22=RHS−m22
LHS/(-2)=RHS/(-2)
Put minus sign in front of fraction
aa=1
Rearrange equation
ℓ1⊥ℓ2⇒m1⋅m2=-1
Substitute expressions
(a)2=a
Add terms
(a−b)2=a2−2ab+b2
LHS−m12=RHS−m12
LHS−m22=RHS−m22
m1m2=-1
-a(-b)=a⋅b
m1⋅m2=-1⇒ℓ1⊥ℓ2
The biconditional statement has been proven.
ℓ1⊥ℓ2⇔m1⋅m2=-1
Consider two lines on the same coordinate plane and their equations in slope-intercept form. Are the lines perpendicular?
Write an equation in slope-intercept form of the line that passes through (6,-1) and is perpendicular to y=-3x+2.
What makes two lines perpendicular?
x=6, y=-1
b1⋅a=ba
Calculate quotient
LHS−2=RHS−2
Rearrange equation
As seen above, the graph of y=31x−3 is perpendicular to the graph of y=-3x+2 and passes through (6,-1).
Zain's class is modeling a neighborhood that is being built outside of town. For extra credit, Zain decides to use the neighborhood's plumbing plan determine where the pipe that connects a new house to the water supply network will be placed. They have the following plan of the network.
As seen above, the graph of y=-2x+5 is perpendicular to the given line and passes through (3,-1). The new pipe is a part of y=-2x+5.
Substitute (0,4) & (6,0)
Subtract terms
Put minus sign in front of fraction
ba=b/2a/2
x=6, y=4
ca⋅b=ca⋅b
Calculate quotient
LHS−9=RHS−9
Rearrange equation
It can be seen that the lines are perpendicular and that y=23x−5 passes through (6,4), which corresponds to the flower beds.
We begin by recalling that parallel lines have the same slope. In order to determine which lines are parallel we will find the slope of each line. We can find the slope of a line by inspecting the coefficient of the x-term. Let's do it!
Line | Slope |
---|---|
y_1= 4x+3 | 4 |
y_2= x+2 | 1 |
y_3=3+ 5x | 5 |
y_4=1 -4x | - 4 |
y_5=4+ 4x | 4 |
y_6= - x+5 | -1 |
In the table above we can see that the lines y_1 and y_5 both have a slope of 4. All other lines have different slopes. Therefore, y_1 and y_5 are the only parallel lines.
Consider the following diagram.
Let's begin by drawing lines L_1 and L_2.
At first glance the lines appear to be parallel. Let's be sure to find the slope of each line. Recall the Slope Formula. m=y_2-y_1/x_2-x_1 We first need to identify the coordinates of the given points. A=(- 5,- 1) B=(4,7) [0.5em] C=(- 4,- 5) D=(7,5) Next, substitute the coordinates of A and B to find the slope m_1 of L_1.
We find the slope m_2 of L_2 in a similar way.
The slopes of L_1 and L_2 are different. Therefore, the lines are not parallel.
Determine whether the following statement is always, sometimes, or never true.
Two lines with negative slopes are perpendicular. |
Let's begin by recalling the relation between the slopes of two perpendicular lines.
Slopes of Perpendicular Lines Theorem |- In a coordinate plane, two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals.
The above theorem states that if non-vertical lines are perpendicular, the product of their slopes is always -1. The given statement tells us that both lines have negative slopes, so their product will be positive. Since the product is positive it cannot be -1, so the statement is never true.
Consider the following diagram.
Let's begin by drawing lines L_1 and L_2.
At first glance the lines appear to be perpendicular. Let's be sure by finding the slope of each line. Recall the Slope Formula. m=y_2-y_1/x_2-x_1 We first need to identify the coordinates of the given points. &A=(- 2,- 1) &&B=(2,1) [0.5em] &C=(1,- 2) &&D=(-1,2) Next, substitute the coordinates of A and B to find the slope m_1 of L_1.
We find the slope m_2 of L_2 in a similar way.
The Slopes of Perpendicular Lines Theorem tells us that two lines are parallel if and only if their slopes are negative reciprocals. Let's check! m_1=1/2 m_2=-2 [0.5em] m_1* m_2=- 1 ✓ We can see that 12 and -2 are negative reciprocals. Therefore, the lines are perpendicular.