**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 j_{1 }and j_{2 }are parallel.

**Problem 4 :**

In the diagram given below,

Line n_{1} has the equation y = -x/3 -1.

Line n_{2} is parallel to the line n_{1} and passes through the point (3, 2).

Write the equation of the line n_{2}.

**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 ?

**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.

**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.

**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.

**Solution :**

m||n ∠1 ≅ ∠2 n||k ∠2 ≅ ∠3 ∠1 ≅ ∠3 m||k |
Given Corresponding angles postulate Given Corresponding angles postulate Transitive property of congruence Corresponding angle converse |

**Problem 3 :**

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

**Solution : **

Line j_{1} has a slope of

m_{2} = 4/2 = 2

Line j_{2} has a slope of

m_{2} = 2/1 = 2

Since the slope of the lines j_{1 }and j_{2 }are equal, the lines j_{1 }and j_{2 }are parallel.

**Problem 4 :**

In the diagram given below,

Line n_{1} has the equation y = -x/3 -1.

Line n_{2} is parallel to the line n_{1} and passes through the point (3, 2).

Write the equation of the line n_{2}.

**Solution : **

The slope of the line n_{1} is -1/3. Because the lines n_{1} and n_{2} are parallel, they have the same slope. So, the slope of the line n_{2} is also -1/3.

Slope-intercept form equation of a line :

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

Because the line n_{2} is passing through (3, 2), substitute aa(x, y) = (3, 2) amd m = -1/3

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

Simplify

2 = -1 + b

Add 1 to both sides.

3 = b

The equation of the required line is

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

y = -x/3 + 3

**Problem 5 :**

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 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 ?

**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

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