Question

find the scale factor from the given vertices.
a: (-4, 2) a: (-1, \frac{1}{2})
b: (0, 5) b: (0, \frac{5}{4})
c: (4, -2) c: (1, \frac{-1}{2})

\bigcirc 4
\bigcirc -4
\bigcirc \frac{1}{4}
\bigcirc \frac{-1}{4}

Explanation:

Step1: Recall Scale Factor Formula

The scale factor \( k \) for a dilation is found by dividing the coordinates of the image vertex by the coordinates of the original vertex. For a point \( (x, y) \) and its image \( (x', y') \), \( k=\frac{x'}{x}=\frac{y'}{y} \) (assuming the dilation is centered at the origin, which is consistent here as the x - coordinates and y - coordinates scale by the same factor for each point).

Step2: Calculate Scale Factor for Point A

For point \( A(-4, 2) \) and its image \( A'(-1,\frac{1}{2}) \):

• For the x - coordinate: \( k_x=\frac{-1}{-4}=\frac{1}{4} \)
• For the y - coordinate: \( k_y=\frac{\frac{1}{2}}{2}=\frac{1}{2}\div2=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4} \)

Step3: Calculate Scale Factor for Point B

For point \( B(0, 5) \) and its image \( B'(0,\frac{5}{4}) \):

• For the x - coordinate: Since \( x = 0 \), we use the y - coordinate. \( k=\frac{\frac{5}{4}}{5}=\frac{5}{4}\div5=\frac{5}{4}\times\frac{1}{5}=\frac{1}{4} \)

Step4: Calculate Scale Factor for Point C

For point \( C(4, - 2) \) and its image \( C'(1,-\frac{1}{2}) \):

• For the x - coordinate: \( k_x=\frac{1}{4} \)
• For the y - coordinate: \( k_y=\frac{-\frac{1}{2}}{-2}=\frac{1}{2}\div2=\frac{1}{4} \)

Answer:

\(\frac{1}{4}\) (corresponding to the option \(\frac{1}{4}\))