Question

Two horses are ready to return to their barn after a long workout session at the track. The horses are at coordinates 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 represents 100 meters. Which horse is closer to the barn? Justify your answer.

Answer 1

$\bf \textit{distance between 2 points}\\ \quad \\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &H({{ 1}}\quad ,&{{ 10}})\quad % (c,d) &B({{ -3}}\quad ,&{{ -9}}) \end{array}\quad % distance value d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2} \\\\\\ HB=\sqrt{(-3-1)^2+(-9-10)^2}\implies HB=\sqrt{(-4)^2+(-19)^2} \\\\\\ HB=\sqrt{377}\implies HB\approx 19.4\cdot \stackrel{meters}{100}\implies HB\approx 1940~m\\\\ -------------------------------$

$\bf \textit{distance between 2 points}\\ \quad \\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &Z({{ 10}}\quad ,&{{ 1}})\quad % (c,d) &B({{ -3}}\quad ,&{{ -9}}) \end{array}\quad % distance value d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2} \\\\\\ ZB=\sqrt{(-3-10)^2+(-9-1)^2}\implies HB=\sqrt{(-13)^2+(-10)^2} \\\\\\ ZB=\sqrt{269}\implies ZB\approx 16.4\cdot \stackrel{meters}{100}\implies ZB\approx 1640~m$

Answer 2

H(1,10)   Z(10,1) B(-3,-9)
1st Calculate the distance H(horse) H to B(barn)
2nd Calculate the distance Z(horse) H to B(barn)
3rd Compare 


1st distance H(horse) H to B(barn) 
HB = √[(x₂-x₂)²+(y₂-y₁)₂] → √[-3-1)² + (-9-10)²]=√377 = 19.42

2nd distance ZB = √[(-3-10)²+(-9-1)²] = √269 = 16.4

HB in meter =19.42 x 100  = 1,942 m
ZB in  meter = 16.4 x 100  =  1,640 m

Obviously HB > ZB  and Z is closer to  the Barn