# Cartesian Plane Worksheet

Cartesian Plane Worksheet
• Page 1
1.
What is the distance of the point (6, 8) from the origin?
 a. 10 b. 10$\sqrt{2}$ c. 9 d. 11

#### Solution:

Distance of (6, 8) from the origin (0, 0) = (0 - 6)2+(0 - 8)2
[Distance formula.]

= 36 + 64

= 100

= 10

2.
What is the distance of the point (6$a$, $a$2 - 9) from the origin?
 a. ($a$2 - 3) units b. ($a$2 - 6) units c. ($a$2 + 9) units d. ($a$2 - 9) units

#### Solution:

Distance of (6a, a2 - 9) from the origin (0, 0) = (0 - 6a)2+[0 - (a2 - 9)]2
[Distance formula.]

= 36a2+(a2 - 9)2

= 36a2+a4 + 81 - 18a2

= a4+18a2+81

= (a2+9)2

= (a2 + 9) units

3.
If the distance from the origin to the point ($k$, $k$ + 2) is 10 units, then find the value of $k$.
 a. -6, 8 b. 6, - 8 c. -6, - 8 d. 6, 8

#### Solution:

The distance from the origin to (k, k + 2) = 10

(k-0)2+(k+2-0)2 = 10
[Distance formula.]

k2 + (k + 2)2 = 100
[Square on both sides.]

k2 + k2 + 4k + 4 = 100

2k2 + 4k - 96 = 0

k2 + 2k - 48 = 0
[Simplify.]

(k + 8)(k - 6) = 0
[Factor.]

k = 6, - 8.
[Solve for k.]

4.
What is the distance between the points (3, 3) and (12, 15)?
 a. 225 units b. 15 units c. 144 units d. 81 units

#### Solution:

The distance between the points (3, 3) and (12, 15) = (12 - 3)2+(15 - 3)2
[Use the distance formula.]

= 81 + 144

= 225

= 15 units.

5.
The distance between the points (4, 4) and (10, $k$) is 10 units. What are the values of $k$?
 a. 12, 4 b. -12, -4 c. 4, -12 d. 12, -4

#### Solution:

The distance between the two points (4, 4) and (10, k) = 10

(10-4)2+(k-4)2 = 10

36 + (k - 4)2 = 100
[Square on both sides.]

(k - 4)2 = 64

k - 4 = 8 , -8

k = 12, -4.
[Solve for k.]

6.
If θ is a real number, then find the distance between the points (8sin 5θ, 8cos 5θ) and (- 8cos 5θ, 8sin 5θ).
 a. 8 b. 8$\sqrt{2}$ units c. 9$\sqrt{2}$ units d. 128

#### Solution:

The distance between the points (8sin 5θ, 8cos 5θ) and (- 8cos 5θ, 8sin 5θ)
= (- 8cos 5θ - 8sin 5θ)2+(8sin 5θ - 8cos 5θ)2

= 2(64cos² 5θ+64sin² 5θ)
[Expand and simplify.]

= 128
[Use sin² 5θ + cos² 5θ = 1.]

= 82 units

7.
Find the distance between the points (7, 0) and (8, tan 7$\theta$) for all real values of $\theta$.
 a. - sec 7$\theta$ b. |sec 7$\theta$ | c. ± sec 7$\theta$ d. sec 7$\theta$

#### Solution:

The distance between the points (7, 0) and (8, tan 7θ) = (8 - 7)2+(tan 7θ - 0)2
[Distance formula.]

= 1+tan2 7θ

= sec2 7θ
[Use 1 + tan2 7θ = sec2 7θ.]

= | sec 7θ |
[Use x2 = | x |.]

8.
The distance of the point (cot 8θ, 9) from (0, 10) is $\sqrt{2}$ units where 8θ is an acute angle. Find the value of θ.
 a. $\frac{\pi }{7}$ b. $\frac{\pi }{9}$ c. $\frac{\pi }{32}$ d. $\frac{\pi }{8}$

#### Solution:

The distance of the point (cot 8θ, 1) from the origin = 2

(0 - cot 8θ)2+(0 - 1)2 = 2
[Distance formula.]

cot2 8θ + 1 = 2
[Square on both sides.]

cot2 8θ = 1

cot 8θ = 1
[Solve for cot 8θ.]

8θ = π4
[Solve for 8θ.]

θ = π32
[Solve for θ.]

9.
If $a$ is any real number, then what is the distance from (4, 0) to (0, $a$)?
 a. > 4 units b. ≥ 4 units c. < 4 units d. = 4 units

#### Solution:

The distance from (4, 0) to (0, a) is (0 - 4)2+(a - 0)2.
[Distance formula.]

= 16+a2

≥ 4 units
[As 16+a2 ≥ 4.]

10.
If A = ($k$, $k$), B = (3 + $k$, 4 + $k$), C = (4 + $k$, 3 + $k$) are any three points of a plane, then for all the real values of $k$ which of the following is correct?
 a. AB < AC b. AB = AC c. 2AB = AC d. AB > AC

#### Solution:

A = (k, k), B = (3 + k, 4 + k), C = (4 + k, 3 + k) where k is a real number.

AB = (3 + k - k)2+(4 + k - k)2 = 9 + 16 = 5
[Distance formula.]

AC = (4 + k - k)2+(3 + k - k)2 = 16 + 9 = 5
[Distance formula.]

So for all real values of k, AB = AC.