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# Distance Formula, Unknown y-coordinate

Q: Find the possible values of a, if the distance between the points is 5, and the coordinates are (1,1) and
(4,a). Thank you.

A: Are you familiar with the distance formula? The distance d, between two points (x1,y1) and (x2,y2) is given by:

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

In your question, $d = 5, x_1 =1, y_1=1, x_2=4, y_2 =a$. Rearrange the distance formula to solve for a:

$5 = \sqrt{(4-1)^2 + (a-1)^2}$

Square both sides to get rid of the square root.

25 = (4-1)^2 + (a-1)^2
25 = 3^2 + (a-1)(a-1)

Do FOIL:
25 = 9 + a^2 -2a + 1
0 = a^2 -2a + 1 + 9 – 25
0 =a^2 -2a -15

Solve by factoring. Think, what are two numbers that multiply to -15 and add up to -2a? The answer? -5 and 3.

0 = (a-5)(a+3)

So either a-5=0, which means a=5, or a+3=0, which means a= -3.

The possible values of a are 5 and -3.