# Distance and Midpoint Formula Worksheet

Distance and Midpoint Formula Worksheet
• Page 1
1.
A person started running accross the central park from 0.6 miles North of 59th street and 0.01 miles East of Central Park West. He stopped running at 1.4 miles North of 59th street and 0.61 miles East of Central Park West. Find the distance he ran.
 a. 1.5 miles b. 2 miles c. 1 mile d. 2.5 miles

#### Solution:

The person started running from 0.6 miles North of 59th street and 0.01 miles East of Central Park West. He stopped at 1.4 miles North of 59th street and 0.61 miles East of Central Park West. Superimpose the locations on a map.

Plot the coordinates for the starting and the end point as shown in the figure.

The coordinates from the map are, (0.01, 0.6) and (0.61, 1.4).

To find the distance he ran, we use the distance formula,
D = (x2 -x1)2 +(y2 -y1)2

= (0.61 - 0.01)2 +(1.4 - 0.6)2
[Sunstitute the values, (x1, y1) = (0.01, 0.6) and (x2, y2) = (0.61, 1.4).]

= (0.6)2 +(0.8)2

=0.36 + 0.64

= 1.00 = 1

Therefore, the person ran for1mile.

2.
The map represents the overview of the few places from the Christine's home. Each side of a square in the cooridnate plane that is superimposed on the map represents 2 miles. Find the distance from Christine's home to the Zoo.

 a. 24 miles b. 17 miles c. 10 miles d. 18 miles

#### Solution:

To find the distance between from home to the Zoo, we use the distance formula,
D = (x2 -x1)2 +(y2 -y1)2

= (8 - 0)2 +(15 - 0)2
[Sunstitute the values, (x1, y1) = (0, 0) and (x2, y2) = (8, 15).]

= 82 +152

=64 + 225

= 289 = 17

Therefore, the distance from Christine's home to the Zoo is 17 miles.

3.
The map represents the overview of the few places from the Rick's home. Each side of a square in the cooridnate plane that is superimposed on the map represents 1 mile. Rick plans to go to beach from the park near his home.Find the distance Rick has to travel, from the park to the beach.

 a. 4 miles b. 10 miles c. 15 miles d. 5 miles

#### Solution:

To find the distance between the park and the beach, we use the distance formula,
D = (x2 -x1)2 +(y2 -y1)2

= (5 - 2)2 +(9 - 5)2
[Substitute the values, (x1, y1) = (2, 5) and (x2, y2) = (5, 9).]

= 32 +42

=9 + 16

= 25 = 5

Therefore, Rick has to travel 5 miles to reach the beach from the park.

4.
Find the perimeter of quadrilateral ABCD.

 a. 28 units b. 24.08 units c. 30 units d. 30.25 units

#### Solution:

Distance between two points (x1, y1) and (x2, y2) = (x2 -x1)² +   (y2 -y1)²
[Formula.]

AB = (11 - 3)² + (8 - 8)²
[Replace (x1, y1) with (3, 8) and (x2, y2) with (11, 8).]

AB = 82 + 0 = 8
[Simplify.]

BC = (8 - 11)² + (2 - 8)²
[Replace (x1, y1) with (11, 8) and (x2, y2) with (8, 2).]

BC = (- 3)² + (- 6)² = 35
[Simplify.]

CD = (3 - 8)² + (4 - 2)²
[Replace (x1, y1) with (8, 2) and (x2, y2) with (3, 4).]

CD = (- 5)²+ 2² = 29
[Simplify.]

DA = (3 - 3)² + (8 - 4)²
[Replace (x1, y1) with (3, 4) and (x2, y2) with (3, 8).]

DA = 0² + 4² = 4
[Simplify.]

Perimeter of ABCD = AB + BC + CD + DA

= 8 + 6.69 + 5.39 + 4 = 24.08
Perimeter = 8 + 35 + 29 + 4
[Substitute the values of AB, BC, CD, DA and add.]

The perimeter of ABCD = 24.08 units.

5.
The endpoints of a line segment are (2, - 4) and (2, - 2). Find the midpoint of the line segment.
 a. (0, - 1) b. (2, - 3) c. (- 4, - 6) d. (2, 1)

#### Solution:

The endpoints of the line segment has coordinates (2, - 4) and (2, - 2).

Midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is [x1+x22, y1+y22]
[Use the midpoint formula.]

Midpoint = [2+2 / 2, - 4 + (- 2)2]
[Substitute the values.]

(4 / 2, - 6 / 2)
[Simplify.]

Midpoint = (2, - 3)
[Write the fractions in simplest form.]

The midpoint of the line segment = (2, - 3)

6.
In a coordinate plane L is at (- 4, - 4) and M is at (- 7, - 7). Find the distance between L and M.
 a. 8 units b. 15.56 units c. 6 units d. 4.24 units

#### Solution:

Distance, d = (x2-x1)2+(y2-y1)2
[Use the distance formula.]

d = [(- 7) - (- 4)]2+[(- 7) - (- 4)]2
[Replace (x2, y2) with (- 7, - 7) and (x1, y1) with (- 4, - 4).]

d = (- 3)2+(- 3)2
[Subtract.]

d = 18
[Simplify.]

d = 4.24
[Find the positive square root.]

So, distance between L and M = 4.24 units.

7.
The distance between A and B is 6 units. Which among the following can be the ordered pair for B, if A is (- 3, 2)?
 a. (4, 2) b. (3, 3) c. (- 3, - 2) d. (3, 2)

#### Solution:

The distance between A and B is 6 units.

Distance, d = (x2-x1)2+(y2-y1)2
[Use the distance formula.]

Distance between (- 3, 2) and (4, 2) is (4 - (- 3))2 +(2 - 2)2 = 72+02 = 49 + 0 = 49 ≠ 6

Distance between (- 3, 2) and (3, 3) is (3 - (- 3))2+(3 - 2)2 = 62+12 = 36 + 1 = 37 ≠ 6

Distance between (- 3, 2) and (- 3, - 2) is ((- 3) - (- 3))2+((- 2) - 2)2 = 02+(- 4)2 = 0+16 = 16 ≠ 6

Distance between (- 3, 2) and (3, 2) is (3 - (- 3))2+(2 - 2)2 = 62+02 = 36+0 = 36 = 6

The coordinates in choices A, B and C are at a distance other than 6 units from the point A.

The point (3, 2) is at a distance of 6 units from A.

So, the coordinates of B are (3, 2).

8.
Find the perimeter of the right triangle shown.

 a. 32.5 units b. 27.4 units c. 26.2 units d. 30.5 units

#### Solution:

From the figure, P is at (- 3, 4), Q is at (6, 4) and R is at (6, - 3).

Distance d = (x2-x1)2+(y2-y1)2

PQ = [6-(- 3)]2+(4-4)2
[Replace (x1, y1) with (- 3, 4) and (x2, y2) with (6, 4).]

PQ = 92+02=81 = 9
[Simplify.]

QR = (6-6)2+(- 3 - 4)2
[Replace (x1, y1) with (6, 4) and (x2, y2) with (6, - 3).]

QR = 02+(- 7)2=0+49=49 = 7
[Simplify.]

RP = (- 3 - 6)2+[4-(- 3)]2
[Replace (x1, y1) with (6, - 3) and (x2, y2) with (- 3, 4).]

RP = (- 9)2+72=81+49=130
[Simplify.]

Perimeter of ΔPQR = PQ + QR + RP = 9 + 7 + 130
[Substitute the values for PQ, QR and RP.]

= 16 + 11.4 = 27.4

The perimeter of ΔPQR is 27.4 units.

9.
Find the midpoint of the line segment PQ which connects the points P(7, 4) and Q(9, 6).
 a. (9, 4) b. (7, 4) c. (10, 9) d. (8, 5)

#### Solution:

Given (x1, y1) and (x2, y2) the Midpoint = (x1+x22, y1+y22)
[Use the midpoint formula.]

Midpoint of PQ = (7+9 / 2, 4+6 / 2)
[Replace (x1, y1) with (7, 4) and (x2, y2) with (9, 6).]

Midpoint = (16 / 2 , 10 / 2 )
[Simplify the numerators.]

= (8, 5)
[Write the fractions in simplest form.]

The coordinates of the midpoint of the line segment PQ = (8, 5).

10.
The distance between A(5, 6) and B(2, $x$) is 3$\sqrt{2}$ units. Find the coordinates of B.
 a. (0 , 3) or (2, 0) b. (2 , 3) or (2, 9) c. (2 , 4) or (1, 9) d. (- 2 , 3) or (2, 9)

#### Solution:

The distance between (x1, y1) and (x2, y2) = d = (x2-x1)2+(y2-y1)2
[Use the distance formula.]

Distance between AB, d = (2-5)2+(x-6)2
[Replace (x1, y1) with (5, 6) and (x2, y2) with (2, x).]

d = (- 3)2+(x-6)2
[Subtract.]

d = 9+(x-6)2
[Simplify.]

d = 9+x2+36-12x
[Apply exponents.]

= x2-12x+45

The distance between A and B is 32 units.

x2-12x+45=32
[Equate distances.]

x2 - 12x + 45 = (32)2 = 18
[Squaring on both sides.]

x2 - 12x + 27 = 0
[Write in general form.]

(x - 3)(x - 9) = 0
[Factor.]

x = 3 or x = 9

The coordinates of B are either (2, 3) or (2, 9)
[Substitute x values.]