# Lines and Angles Worksheets

Lines and Angles Worksheets
• Page 1
1.
Find the inclination of the line $y$ = $x$ + 32.
 a. 90o b. 45o c. 60o d. 30o

#### Solution:

y = x + 32

Slope = m = tan θ = 1
[Compare with y = mx + c.]

θ = tan-1(1) = 45o

So, the inclination of the line is θ = 45o

2.
What is the measure of the angle between the two lines having the slopes - 7 and $\frac{1}{7}$.
 a. 45o b. 60o c. 30o d. 90o

#### Solution:

The slopes of the lines are m1= - 7, m2 = 1 / 7.

The angle between the two lines is θ = Tan-1[-7-171-7 17] .
[Use the formula to find the angle between the two lines.]

= Tan-1()

= 90o

3.
If a non vertical line has an inclination θ with the positive $x$-axis, then find the slope $m$ of the line in terms of θ.
 a. cosec $\theta$ b. cot $\theta$ c. tan $\theta$ d. sin $\theta$

#### Solution:

If a non vertical line has an inclination θ with the positive x-axis, then the slope m of the line is given by tan θ.

4.
Find the measure of the angle between two lines having slopes 7 and $\frac{1}{7}$ .
 a. tan-1(48) b. tan-1(49) c. tan-1($\frac{24}{7}$) d. tan-1($\frac{1}{14}$)

#### Solution:

The slopes of the lines are m1 = 7, m2 = 1 / 7

The angle between the two lines is θ = Tan-1[7-171+7.17] .
[Use the formula to find the angle between two lines.]

= Tan-1[(49-17)2]

= tan-1(24 / 7)

5.
Find the measure of the angle between the two lines 8$x$ + 9$y$ = 1 and 4$x$ + 7$y$ = 1.
 a. - tan-1($\frac{4}{19}$) b. tan-1($\frac{1}{95}$ ) c. tan-1(20 ) d. tan-1(7 )

#### Solution:

The two lines are 8x + 9y - 1 = 0 and 4x + 7y - 1 = 0.

y = (- 8 / 9)x + (1 / 9) and y = (- 4 / 7)x + (1 / 7)
[Convert into y = mx + c form.]

The slopes of the lines are m1 = - 8 / 9 and m2 = - 4 / 7
[Write the slopes of the two lines.]

The angle between the given lines is θ = Tan-1[-89+471+(-89)(-47)] .
[Use the formula to find the angle between the lines.]

= - tan-1(4 / 19)

6.
What is the measure of the angle between the two lines 5$x$ + $y$ + 4 = 0 and 5$x$ - $y$ - 3 = 0?
 a. tan-1(24 ) b. - tan-1($\frac{5}{12}$) c. tan-1($\frac{5}{12}$) d. tan-1(10 )

#### Solution:

The two lines are 5x + y + 4 = 0 and 5x - y - 3 = 0.

y = - 5x - 4 and y = 5x - 3
[Convert it into y = mx + c form.]

The slopes of the two lines are m1 = - 5 and m2 = 5
[Write the slopes of the two lines.]

The angle between the given lines is θ = Tan-1[-5 - 51 - 25]
[Use the formula to find the angle between the two lines.]

= tan-1(5 / 12)

7.
What is the perpendicular distance between the point (2, 2) and the line 12$x$ + 16$y$ + 2 = 0.
 a. 20 units b. $\frac{1}{20}$ units c. 58 units d. $\frac{29}{10}$units

#### Solution:

The perpendicular distance between the point (2, 2) and the line 12x + 16y + 2 = 0 is d = |12(2)+16(2)+2|122+162
[The perpendicular distance from the point P (x1, y1) and the line Ax + By + C = 0 is d = |Ax1+By1+C|A2+B2.]

= 29 / 10 units

8.
What is the slope of the line on which for every point the ordinate is same?
 a. Undefined b. Infinitely large c. Infinitely small

#### Solution:

Let L be the line on which for every point the ordinate is same.

So, let P(14,9) and Q(19,9) be the two points on line L.

Slope of line L = Slope of line joining P,Q = m = ΔyΔx

= 9 - 919 - 14
[Use the slope formula.]

= 05 = 0

9.
The angle of inclination of a line is . What is the slope of the line?
 a. -1 b. 1 c. -2 d. 2

#### Solution:

The inclination of the line = θ = 3 π4

Slope of the line = m = tan θ
[Use the definition of slope.]

= tan 3 π4 = -1

10.
What is the slope of a line on which for every point the $x$ coordinate is same?
 a. Infinitely small b. Undefined c. Infinitely large

#### Solution:

Let L be the line on which for every point the x coordinate is same.

So, let A(3, 6), B(3, 9) be two points on line L.

Slope of line L = Slope of line joining A,B = m = ΔyΔx
[Use the slope formula.]

= 9-63 - 3

= 30 = Undefined
[Division by 0 is not defined]