7.3 Applications of Coordinate Geometry in Real life Situations

7.3 Applications of Coordinate Geometry in Real life Situations

Example 14

On a map, Town A is at coordinates $(2, 3)$ and Town B is at $(-4, -1)$. What is the distance between the two towns?

Solution: Use the distance formula:

Substitute the values:

Thus, the distance between Town A and Town B is approximately $7.21$ units.

Example 15

Suppose two cities, City A and City B, are represented by the coordinates $(3, 4)$ and $(7, 1)$ on a map. Find the straight-line distance between the two cities.

Solution: We apply the distance formula:

Thus, the straight line distance between City A and City B is $5$ units.

Example 16

An Engineer is building a bridge between two points on a riverbank. Suppose the coordinates of the two points where the bridge will start and end are $(2, 5)$ and $(8, 9)$. Find the coordinates of the midpoint, which will represent the centre of the bridge.

Solution: We apply the midpoint formula:

Thus, the centre of the bridge is at the point $(5, 7)$

Example 17

A landscaper is designing a triangular garden with corners at points $A(2, 3)$, $B(5, 7)$, and $C(6, 2)$. Calculate the lengths of the sides of the triangular garden.

Solution: Use the distance formula to find the length of each side:

$|AB| = \sqrt{(5 - 2)^2 + (7 - 3)^2}$
$|AB| = \sqrt{(3)^2 + (4)^2}$
$|AB| = \sqrt{9 + 16} = \sqrt{25} = 5$ units

$|BC| = \sqrt{(6 - 5)^2 + (2 - 7)^2}$
$|BC| = \sqrt{(1)^2 + (-5)^2}$
$|BC| = \sqrt{1 + 25} = \sqrt{26} \approx 5.10$ units

$|AC| = \sqrt{(6 - 2)^2 + (2 - 3)^2}$
$|AC| = \sqrt{(4)^2 + (-1)^2}$
$|AC| = \sqrt{16 + 1} = \sqrt{17} \approx 4.12$ units

Thus, the lengths of the sides of the triangular garden are:
$mAB = 5$ units, $mBC \approx 5.10$ units, $mAC \approx 4.12$ units

Example 18

A pilot needs to travel from city A$(50, 60)$ to city B$(120, 150)$. Determine the heading angle the plane should take relative to the east direction.

Solution: The heading angle can be calculated using the slope:

Let $\theta$ be the required angle, then

Thus, the plane should take a heading angle of $52.13^\circ$ north of east.

lightbulb

Do you know?
Aviation is the operation and flight of aircraft, including airplanes, helicopters and drones.
Navigation is the process of determining and controlling the route of a vehicle from one place to another.

Latitude measures how far a location is from the equator. It ranges from $0^\circ$ at the equator to $90^\circ$ north (at the North Pole) or $90^\circ$ south (at the South Pole).
Longitude measures how far a location is from the Prime Meridian (which runs through Greenwich, London). It ranges from $0^\circ$ at the Prime Meridian to $180^\circ$ east and $180^\circ$ west.

Example 19

Abdul Hadi is traveling from point A (Latitude $10^\circ$ N, Longitude $50^\circ$ E) to point B (Latitude $20^\circ$ N, Longitude $60^\circ$ E). Find the midpoint of his journey in terms of latitude and longitude.

Solution:
Given that
Point A (Latitude $10^\circ$ N, Longitude $50^\circ$ E)
Point B (Latitude $20^\circ$ N, Longitude $60^\circ$ E)

Midpoint latitude $= \dfrac{10^\circ + 20^\circ}{2} = 15^\circ N$
Midpoint longitude $= \dfrac{50^\circ + 60^\circ}{2} = 55^\circ E$

Thus, the midpoint of Abdul Hadi's journey would be at (Latitude $15^\circ$ N, Longitude $55^\circ$ E).

Example 20

A landscaper is designing a straight pathway from P$(2, 3)$ to Q$(8, 9)$. What is the length of the pathway?

Solution:
The length of the straight pathway can be found using the distance formula:

Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$= \sqrt{(8 - 2)^2 + (9 - 3)^2}$
$= \sqrt{(6)^2 + (6)^2}$
$= \sqrt{36 + 36}$
$= 6\sqrt{2}$

So, the length of the pathway is approximately $6\sqrt{2}$ units.