Question

the right triangle on the right is a scaled copy of the right triangle on the left. identify the scale factor. express your answer as a whole number or fraction in simplest form.

Explanation:

Step1: Determine the scale - factor formula

The scale factor \(k\) of a scaled copy is given by \(k=\frac{\text{length of a side in the scaled copy}}{\text{length of the corresponding side in the original}}\).

Step2: Identify corresponding sides

Let's take the vertical sides. The length of the vertical side in the original triangle is \(3\), and the length of the vertical side in the scaled - copy is \(\frac{4}{3}\). Or we can take the horizontal sides: the length of the horizontal side in the original triangle is \(4\), and the length of the horizontal side in the scaled - copy is \(1\). Using the horizontal sides: \(k = \frac{1}{4}\times\frac{4}{1}=\frac{1}{4}\) (using \(k=\frac{\text{scaled side}}{\text{original side}}\), \(k=\frac{1}{4}\) when considering the horizontal sides. Using the vertical sides: \(k=\frac{\frac{4}{3}}{3}=\frac{4}{9}\) (incorrect). Using the formula with the correct pair:
If we use the ratio of the sides of the scaled copy to the original. Let the original triangle have sides \(a = 3\) (vertical) and \(b = 4\) (horizontal), and the scaled - copy have sides \(a'=\frac{4}{3}\) (vertical) and \(b' = 1\) (horizontal).
We know that \(k=\frac{b'}{b}\) (since they are corresponding sides).
Substitute \(b'=1\) and \(b = 4\) into the formula \(k=\frac{b'}{b}\).
\(k=\frac{1}{4}\)

Answer:

\(\frac{1}{4}\)