How do you find the value of a given the points (2,a), (2,3) with a distance of 10?

The formula for calculating the distance between two points is:

#d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)#

Substituting the values given in the problem for the distance and the points gives:

#10 = sqrt((color(red)(2) - color(blue)(2))^2 + (color(red)(3) - color(blue)(a))^2)#

We can now solve for #a#:

#10 = sqrt(0^2 + (color(red)(3) - color(blue)(a))^2)#

#10 = sqrt(0 + (color(red)(3) - color(blue)(a))^2)#

#10 = sqrt((color(red)(3) - color(blue)(a))^2)#

Remember, taking the square root results in a positive and negative result:

#10 = +-(color(red)(3) - color(blue)(a))#

Solution 1)

#10 = +(color(red)(3) - color(blue)(a))#

#10 = color(red)(3) - color(blue)(a)#

#-3 + 10 = -3 + color(red)(3) - color(blue)(a)#

#7 = 0 - color(blue)(a)#

#7 = -color(blue)(a)#

#-1 * 7 = -1 * -color(blue)(a)#

#-7 = a#

#a = -7#

Solution 2)

#10 = -(color(red)(3) - color(blue)(a))#

#10 = -color(red)(3) + color(blue)(a)#

#3 + 10 = 3 - color(red)(3) + color(blue)(a)#

#13 = 0 + color(blue)(a)#

#13 = color(blue)(a)#

#a# can equal either #-7# or #13#