# PARALLEL AND PERPENDICULAR LINES WORKSHEET

Problem 1 :

Think of each segment in the diagram as part of a line. Which of the lines appear to fit the descriptions given below?

(i) Parallel to AB and contains D

(ii) Perpendicular to AB and contains D

(iii) Skew to AB and contains D

(iv) Name the plane(s) that contain D and appear to be parallel to plane ABE.

Problem 2 :

In the diagram given below, lines m, n and k represent three of the oars. If m||n and n||k, then prove m||k. Problem 3 :

In the diagram given below, find the slope of each line. Determine whether the lines jand jare parallel. Problem 4 :

In the diagram given below, line n1 has the equation

y = -x/3 -1

Line n2 is parallel to the line n1 and passes through the point (3, 2).

Write the equation of the line n2. Problem 5 :

Decide whether the lines are perpendicular.

Line 1 : y = 3x/4 + 2

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

Problem 6 :

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?   Which of the lines appear to fit the descriptions given below?

(i) Parallel to AB and contains D

(ii) Perpendicular to AB and contains D

(iii) Skew to AB and contains D

(iv) Name the plane(s) that contain D and appear to be parallel to plane ABE.

Solution (i) :

CD, GH and EF are all parallel to AB. But, only CD passes through D and is parallel to AB.

Solution (ii) :

BC, AD, AE and BF are all perpendicular to AB. But, only AD passes through D and is perpendicular to AB.

Solution (iii) :

DG, DH and DE all pass through D and are skew to AB.

Solution (iv) :

Only plane DCH contains D and is parallel to plane ABE. Statementsm||n∠1 ≅ ∠2n||k∠2 ≅ ∠3∠1 ≅ ∠3m||k ReasonsGivenCorresponding angles postulateGivenCorresponding angles postulateTransitive property of congruenceCorresponding angle converse Line j1 has a slope of

m2 = 4/2 = 2

Line j2 has a slope of

m2 = 2/1 = 2

Since the slope of the lines jand jare equal, the lines jand jare parallel. The slope of the line n1 is -1/3. Because the lines n1 and n2 are parallel, they have the same slope. So, the slope of the line n2 is also -1/3.

Slope-intercept form equation of a line :

y = mx + b ----(1)

Because the line n2 is passing through (3, 2), substitute (x, y) = (3, 2) amd m = -1/3.

2 = (-1/3)(3) + b

2 = -1 + b

3 = b

The equation of the required line is

(1)----> y = (-1/3)x + 3

y = -x/3 + 3

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. 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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