Exercise 7.3
Exercise 7.3 Solutions
Real-world word problems applying distance formula, midpoint formula, and perimeter calculations on coordinate grids.
Question 1
Houses Distance
If the houses of two friends are represented by coordinates (2, 6) and (9, 12) on a grid. Find the straight line distance between their houses if the grid units represent kilometers.
Solution Step-by-Step
Let House A be at (2, 6) and House B be at (9, 12).
Using the distance formula:
Using the distance formula:
d
=
√(x2 - x1)2 + (y2 - y1)2
|AB|
=
√(9 - 2)2 + (12 - 6)2
=
√(7)2 + (6)2
=
√49 + 36 = √85 ≈ 9.22 km
Conclusion: The straight-line distance between their houses is approximately 9.22 km.
Question 2
Trail Midpoint
Consider a straight trail (represented on a coordinate plane) that starts at point (5, 7) and ends at point (15, 3). What are the coordinates of the midpoint?
Solution Step-by-Step
Let the starting point be A(5, 7) and ending point be B(15, 3).
Using the midpoint formula:
Using the midpoint formula:
M(xm, ym)
=
M(
x1 + x22
, y1 + y22
)
M
=
M(
Conclusion: The coordinates of the midpoint of the trail are M(10, 5).
5 + 152
, 7 + 32
) = M(202
, 102
) = M(10, 5)
Question 3
Buildings Distance
An architect is designing a park with two buildings located at (10, 8) and (4, 3) on the grid. Calculate the straight-line distance between the buildings. Assume the coordinates are in meters.
Solution Step-by-Step
Let the buildings be at A(10, 8) and B(4, 3).
Using the distance formula:
Using the distance formula:
|AB|
=
√(4 - 10)2 + (3 - 8)2
=
√(-6)2 + (-5)2
=
√36 + 25 = √61 ≈ 7.81 meters
Conclusion: The distance between the buildings is approximately 7.81 meters.
Question 4
Delivery Distance
A delivery driver needs to calculate the distance between two delivery locations. One location is at (7, 2) and the other is at (12, 10) on the city grid map, where each unit represents kilometers. What is the distance between the two locations?
Solution Step-by-Step
Let locations be A(7, 2) and B(12, 10).
Using the distance formula:
Using the distance formula:
|AB|
=
√(12 - 7)2 + (10 - 2)2
=
√52 + 82
=
√25 + 64 = √89 ≈ 9.43 km
Conclusion: The distance between the two delivery locations is approximately 9.43 km.
Question 5
Race Midpoint
The start and end points of a racetrack are given by coordinates (3, 9) and (9, 13). What is the midpoint of the track?
Solution Step-by-Step
Let starting point be A(3, 9) and ending point be B(9, 13).
Using the midpoint formula:
Using the midpoint formula:
M
=
M(
Conclusion: The coordinates of the midpoint of the racetrack are M(6, 11).
3 + 92
, 9 + 132
) = M(122
, 222
) = M(6, 11)
Question 6
Road Midpoint
The coordinates of two points on a road are A(3, 4) and B(7, 10). Find the midpoint of the road.
Solution Step-by-Step
Using the midpoint formula for points A(3, 4) and B(7, 10):
M
=
M(
Conclusion: The coordinates of the midpoint of the road segment are M(5, 7).
3 + 72
, 4 + 102
) = M(102
, 142
) = M(5, 7)
Question 7
Port Navigation
A ship is navigating from port A located at (12°N, 65°W) to port B at (20°N, 45°W). If the ship travels along the shortest path on the surface of the Earth, calculate the straight line distance between the points.
Solution Step-by-Step
Representing ports on a Cartesian grid: Port A is at (12, 65) and Port B is at (20, 45).
Using the distance formula:
Using the distance formula:
|AB|
=
√(20 - 12)2 + (45 - 65)2
=
√(8)2 + (-20)2
=
√64 + 400 = √464 ≈ 21.54 units
Conclusion: The straight line grid distance is approximately 21.54 units (degrees).
Question 8
Field Perimeter
Farah is fencing around a rectangular field with corners at (0,0), (0,5), (8,5) and (8,0). How much fencing material will she need to cover the entire perimeter of the field?
Solution Step-by-Step
Let the corners be A(0,0), B(0,5), C(8,5), and D(8,0).
Calculate the dimensions:
Calculate the dimensions:
Length L (AB)
=
√(0 - 0)2 + (5 - 0)2 = 5 units
Width W (BC)
=
√(8 - 0)2 + (5 - 5)2 = 8 units
Calculate the perimeter:
Perimeter P
=
2(L + W)
=
2(5 + 8) = 2(13) = 26 units
Conclusion: Farah will need 26 units of fencing material.
Question 9
City Distance
An airplane is flying from city X at (40°N, 100°W) to city Y at (50°N, 80°W). Use coordinate geometry to calculate the shortest distance between these cities on the grid.
Solution Step-by-Step
Representing cities on a grid: X(40, 100) and Y(50, 80).
Using the distance formula:
Using the distance formula:
|XY|
=
√(50 - 40)2 + (80 - 100)2
=
√(10)2 + (-20)2
=
√100 + 400 = √500 = 10√5 ≈ 22.36 units
Conclusion: The straight line grid distance between the cities is approximately 22.36 units.
Question 10
Land Plot
A land surveyor is marking out a rectangular plot of land with corners at (3,1), (3,6), (8,6), and (8,1). Calculate the perimeter.
Solution Step-by-Step
Let the corners be A(3,1), B(3,6), C(8,6), and D(8,1).
Calculate dimensions:
Calculate dimensions:
Length L (AB)
=
√(3 - 3)2 + (6 - 1)2 = 5 units
Width W (BC)
=
√(8 - 3)2 + (6 - 6)2 = 5 units
Since length equals width, the plot is a square.
Calculate the perimeter:
Perimeter P
=
4 × Side = 4 × 5 = 20 units
Conclusion: The perimeter of the plot is 20 units.
Question 11
Garden Fencing
A landscaper needs to install a fence around a rectangular garden. The garden has its corners at coordinates A(0,0), B(5,0), C(5,3), and D(0,3). How much fencing is required?
Solution Step-by-Step
Calculate dimensions:
Length L (AB)
=
√(5 - 0)2 + (0 - 0)2 = 5 units
Width W (BC)
=
√(5 - 5)2 + (3 - 0)2 = 3 units
Calculate perimeter:
Perimeter P
=
2(L + W) = 2(5 + 3) = 2(8) = 16 units
Conclusion: The amount of fencing required is 16 units.
Interactive Sandbox
Map Route & Perimeter Planner
Select a scenario, input coordinates, and visualize the route distance, midpoints, or perimeter fencing.
Click on the grid to change Endpoint B