**Perpendicular lines in the coordinate plane worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on perpendicular lines in the coordinate plane.

**Problem 1 :**

In the diagram given below, find the slope of each line. Determine whether the lines j_{1 }and j_{2 }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 ?

**Problem 1 :**

In the diagram given below, find the slope of each line. Determine whether the lines j_{1 }and j_{2 }are perpendicular.

**Solution : **

**Part 1 :**

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

Let (x_{1}, y_{1}) = (0, 3) and (x_{2}, y_{2}) = (3, 1)

Slope (j_{1}) = (y_{2} - y_{1}) / (x_{2} - x_{1})

Slope (j_{1}) = (1 - 3) / (3 - 0)

Slope (j_{1}) = - 2 / 3

**Part 2 :**

Find the slope of the line j_{2}. Line j_{2} is passing through the points (0, 3) and (-4, -3).

Let (x_{1}, y_{1}) = (0, 3) and (x_{2}, y_{2}) = (-4, -3)

Slope (j_{2}) = (-3 - 3) / (-4 - 0)

Slope (j_{2}) = (-6) / (-4)

Slope (j_{2}) = 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 }j_{2 }is -1, the lines j_{1 and }j_{2} are perpendicular.

**Problem 2 :**

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

**Solution : **

**Part 1 :**

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

Let (x_{1}, y_{1}) = (1, -4) and (x_{2}, y_{2}) = (4, 2)

Slope (AC) = (y_{2} - y_{1}) / (x_{2} - x_{1})

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

Slope (AC) = (2 + 4) / 3

Slope (AC) = 6 / 3

Slope (AC) = 2

**Part 2 :**

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

Let (x_{1}, y_{1}) = (-1, 2) and (x_{2}, y_{2}) = (5, -1)

Slope (BD) = (-1 - 2) / [(5 - (-1)]

Slope (BD) = (-3) / 6

Slope (BD) = -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.

**Problem 3 :**

Decide whether the lines are perpendicular.

Line 1 : y = 3x/4 + 2

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

**Solution : **

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.

**Problem 4 :**

Decide whether the lines are perpendicular.

Line 1 : 4x + 5y = 2

Line 2 : 5x + 4y = 3

**Solution : **

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

4x + 5y = 2 5y = -4x + 2 y = -4x/5 + 2/5 Slope = -4/5 |
5x + 4y = 3 4y = -5x + 3 y = -5x/4 + 3/4 Slope = -5/4 |

**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.

**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 ?

**Solution : **

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

After having gone through the stuff given above, we hope that the students would have understood "Perpendicular lines in the coordinate plane worksheet".

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