Question

Two horses re ready to return to their barn after a long workout session at the track. The horses are at coordinate H (1, 10) and Z (10, 1). Their barns are located in the same building, which is at coordinates B (-3, -9). Each unit/grid on the coordinate plane represent 100 meters. Which horse is closer to the barn? justify your answer.

Answer 1

$\text {Distance between two points = }\sqrt{(X_1 - X_2)^2 + (Y_1 - Y_2)^2 }$

Distance between H and B:

$\text {Distance = }\sqrt{(1 + 3)^2 + (10 + 9)^2 }$

$\text {Distance = }\sqrt{ 16 + 361 }$

$\text {Distance = }\sqrt{ 377 }$

$\text {Distance = } 19.4 \text { units}$

Find the distance:

Distance = 19.4 x 100 = 1940 m

Distance between Z and B:

$\text {Distance = }\sqrt{(10 + 3)^2 + (1 + 9)^2 }$

$\text {Distance = }\sqrt{ 169 + 100}$

$\text {Distance = }\sqrt{ 269 }$

$\text {Distance = } 16.4 \text { units}$

Find the distance:

Distance = 16.4 x 100 = 1640 m

Comparing 1940 m and 1640 m:

1940 > 1640

⇒ The horse at coordinate Z(10, 1) is nearer to the barn.

Answer 2

The given information are the coordinates of points H, Z and B. So, the first step to do is to plot these points on a Cartesian plane as shown in the attached picture. We can deduce visually that horse Z is closer to the barns than horse H. But, to further justify the answer, we have to provide the magnitude of the distance of horses H and Z to barn B. In this approach, we use the distance formula:

d = √(x₂ - x₁)² + (y₂ - y₁)²

Use the coordinates of the two points to know their linear distances. These are represented by the red and green lines for each of the horses.

Distance between H and B = √(⁻3 - 1)² + (⁻9-10)²

Distance between H and B = √(⁻3 - 1)² + (⁻9-10)²

Distance between H and B  = 19.4165

Since the scale is 1 unit = 100 meters, the actual distance between horse H and barn B is 19.416*100 = 1,941.65 meters

Distance between Z and B = √(⁻3 - 10)² + (⁻9-1)²

Distance between Z and B = √(⁻3 - 10)² + (⁻9-1)²

Distance between Z and B  = 16.4012

Since the scale is 1 unit = 100 meters, the actual distance between horse Z and barn B is 16.4012*100 = 1,640.12 meters

Comparing the distances: 1,941.65 meters > 1,640.12 meters. Therefore, it justifies that horse Z is nearer to the barn.