Question

△xyz is the image of △xyz under a dilation through point c. xz = 20 and xz = 24. (if the answer has a negative, that means reflection as well.) what scale factor was used in the dilation? 6/5 -6/5 -5/6 5/6

Explanation:

Step1: Recall the formula for scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of image segment}}{\text{length of pre - image segment}}\).

Step2: Substitute the given values

We are given that \(XZ = 20\) (pre - image segment) and \(X'Z'=24\) (image segment). So, \(k=\frac{X'Z'}{XZ}\).
Substituting the values: \(k = \frac{24}{20}\).

Step3: Simplify the fraction

Simplify \(\frac{24}{20}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4. \(\frac{24\div4}{20\div4}=\frac{6}{5}\).
Since the segments \(XZ\) and \(X'Z'\) are in the same direction (no reflection implied by the sign of the scale factor formula result when considering dilation from a point), we don't have a negative sign.

Answer:

\(\frac{6}{5}\) (corresponds to the first option \(6/5\))