What is the distance between #(1,-4)# and #(7,5)#?

3 Answers
May 18, 2018

Answer:

#3sqrt13# or 10.81665383

Explanation:

make a right angle triangle with the two points being the end points of the hypotenuse.

The distance between the #x# values is 7-1=6

The distance between the #y# values is 5- -4=5+4=9

So our triangle has two shorter sides 6 and 9 and we need to find the length of the hypotenuse, use Pythagoras.

#6^2+9^2=h^2#

#36+81+117#

#h=sqrt117=3sqrt13#

May 18, 2018

Answer:

#sqrt117~~10.82" to 2 dec. places"#

Explanation:

#"calculate the distance d using the "color(blue)"distance formula"#

#•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#"let "(x_1,y_1)=(1,-4)" and "(x_2,y_2)=(7,5)#

#d=sqrt((7-1)^2+(5-(-4))^2)#

#color(white)(d)=sqrt(6^2+9^2)=sqrt(36+81)=sqrt117~~10.82#

May 18, 2018

Answer:

#root()117#

Explanation:

If you were to draw a right triangle so that the hypotenuse is the line between #(1,-4)# and #(7,5)#, you would observe that the two legs of the triangle would be of length #6# (i.e. the distance between #x=7# and #x=1#) and #9# (i.e. the distance between #y=5# and #y=-4#). By applying the pythagorean theorem,

#a^2+b^2=c^2#,

where #a # and #b# are the lengths of the legs of a right triangle and #c# is the length of the hypotenuse, we obtain:

#6^2 + 9^2 = c^2#.

Solving for the length of the hypotenuse (i.e. the distance between the points #(1,-4)# and #(7,5)#), we get:

#c=root()117#.

The process of finding the distance between two points by use of a right triangle can be formulated thusly:

Distance# = root()((x_2−x_1)^2+(y_2−y_1)^2)#.

This is called the distance formula, and can be used to expedite the solving of this sort of problem.