PERPENDICULAR LINES IN THE COORDINATE PLANE WORKSHEET

Problem 1 :

In the diagram given below, find the slope of each line. Determine whether the lines j₁ and j₂ are perpendicular.

Problem 2 :

In the diagram given below, find the slope of each line. Determine whether the lines are perpendicular.

Problem 3 :

Decide whether the lines are perpendicular.

Line 1 : y = 3x/4 + 2

Line 2 : y = -4x/3 - 3

Problem 4 :

Decide whether the lines are perpendicular.

Line 1 : 4x + 5y = 2

Line 2 : 5x + 4y = 3

Problem 5 :

In the diagram given below, the equation y = 3x/2 + 3 represents a mirror. A ray of light hits the mirror at (-2, 0). What is the equation of the line p that is perpendicular to the mirror at this point ?

1. Answer :

Part 1 :

Find the slope of the line j₁. Line j₁ is passing through the points (0, 3) and (3, 1).

Let (x₁, y₁) = (0, 3) and (x₂, y₂) = (3, 1).

Slope (j₁) = (y₂ - y₁)/(x₂ - x₁)

Slope (j₁) = (1 - 3)/(3 - 0)

Slope (j₁) = -2/3

Part 2 :

Find the slope of the line j₂. Line j₂ is passing through the points (0, 3) and (-4, -3).

Let (x₁, y₁) = (0, 3) and (x₂, y₂) = (-4, -3).

Slope (j₂) = (-3 - 3)/(-4 - 0)

Slope (j₂) = -6/(-4)

Slope (j₂) = 3/2

Multiply the slopes :

The product is

= (-2/3) x (3/2) 

= -1

Since the product of slopes of the lines j_{1 and} j₂ is -1, the lines  j_1 and j₂ are perpendicular.

2. Answer :

Part 1 :

Find the slope of the line AC. Line AC is passing through the points (1, -4) and (4, 2).

Let (x₁, y₁) = (1, -4) and (x₂, y₂) = (4, 2).

Slope of AC = (y₂ - y₁)/(x₂ - x₁)

= [(2 - (-4)]/(4 - 1)

= (2 + 4)/3

= 6/3

= 2

Part 2 :

Find the slope of the line BD. Line BD is passing through the points (-1, 2) and (5, -1).

Let (x₁, y₁) = (-1, 2) and (x₂, y₂) = (5, -1).

Slope of BD = (-1 - 2)/[(5 - (-1)]

= -3/6

= -1/2

Multiply the slopes :

The product is

= 2 x (-1/2) 

= -1

Since the product of slopes of the lines is -1, the lines  AC and BD are perpendicular.

3. Answer :

When we compare the given equations to slope intercept equation of a line y = mx + b, we get

slope of line 1 = 3/4

slope of line 2 = -4/3

Multiply the slopes :

The product is

= (3/4) x (-4/3) 

= -1

Since the product of slopes of the lines is -1, the given lines are perpendicular.

4. Answer :

Rewrite each equation in slope-intercept form to find the slope. 

Multiply the slopes :

The product is

= (-4/5) x (-5/4)

= 1

Since the product of slopes of the lines is not -1, the given lines  are not perpendicular.

5. Answer :

The slope of the mirror is 3/2. So, the slope of the line p is -2/3.

Let y = mx + b be the equation of the line p.

Substitute (x, y) = (-2, 0) and m = -2/3 to find the value of b.

0 = (-2/3)(-2) + b

0 = 4/3 + b

Subtract 4/3 from both sides.

-4/3 = b

So, the equation of the line p is

y = -2x/3 - 4/3

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