Question

The endpoints of AB are A(9,4) and B(5,-4). The endpoints of its image after a dilation are A'(6,3) and B'(3,-3). Find the scale factor and explain each of your steps. This is for Geometry A. Please help!

Answer 1

The scale factor is equal to
`(4)/(3)`.

Explanation:

The formula to calculate vector length if the following, where
`| |` denotes the length of a vector:

`|\stackrel{\rightarrow}{AB}| = \alpha |\stackrel{\rightarrow}{A'B'}|`

`\stackrel{\rightarrow}{AB} = \left(\begin{array}{c}9 - 5&\\4 - (-4)&\end{array}\right) = \left(\begin{array}{c}4&\\8&\end{array}\right)\\ \\ |\stackrel{\rightarrow}{AB}| = √(4^2 + 8^2) = √(16+64) = √(80)`

`\stackrel{\rightarrow}{A'B'} = \left(\begin{array}{c}6 - 3&\\3 - (-3)&\end{array}\right) = \left(\begin{array}{c}3&\\6&\end{array}\right)\\ \\ |\stackrel{\rightarrow}{AB}| = √(3^2 + 6^2) = √(9+36) = √(45)`

The scale factor formula is then defined by

`|\stackrel{\rightarrow}{AB}| = \alpha |\stackrel{\rightarrow}{A'B'}|\\ \\ \alpha = \frac{|\stackrel{\rightarrow}{AB}|}{|\stackrel{\rightarrow}{A'B'}|}\\ \\ \alpha = (√(80))/(√(45)) = \sqrt{(80)/(45)} = \sqrt{(16)/(9)} = (4)/(3)`