The endpoints of line segment AB are A (9,4) and B (5,-4). Then endpoints of its image after dilation are A' (6,3) and B' (5,-4). Find the scale factor and explain

To find our scale factor, let's determine the length of AB, and compare it to the length of A'B'.

The length of AB is the distance from (9,4) to (5,-4). 
Let's apply the distance formula:
AB = √ ((9-5)² + (4+4)²)
AB = √ (4² + 8²)
AB = √(16 + 64)
AB = √ (80)
AB = 4 √5
The length of AB is 4 √5.
Now for A'B':
A'B' = √ ((6-5)² + (3+4)²)
A'B' = √( 1 + 49)
A'B' = √ 50
A'B' = 5 √2
To find our scale factor k:
A'B' = k AB
5 √2 = 4 √5 * k Divide both sides by 4√5
5 √2 / (4 √5) = k                 To simplify, multiply the left side by (√5 / √5)
(5 *√5 * √2) / (4 *5) = k      the 5's in the numerator and denominator cancel
√5 * √2 / 4 = k                    
√10 / 4 = k

To confirm, make sure AB * k = A'B', using the values that we've calculated.