Question

1. the polygons shown below are similar. find the scale factor and the val

△abc∼△def
written as a fraction, the scale factor =
x =

Explanation:

Step1: Find the scale factor

Since \(\triangle ABC\sim\triangle DEF\), the scale factor \(k\) is the ratio of corresponding sides. Let's take \(AB = 21\) and \(DF=12\). The scale factor \(k=\frac{AB}{DF}=\frac{21}{12}=\frac{7}{4}\) (simplified by dividing numerator and denominator by 3).

Step2: Set up proportion for \(x\)

Corresponding sides \(BC = 3x - 9\) and \(EF = 32\). Using the scale factor \(\frac{7}{4}\), we have \(\frac{BC}{EF}=\frac{7}{4}\). Substitute \(BC = 3x - 9\) and \(EF = 32\) into the proportion: \(\frac{3x - 9}{32}=\frac{7}{4}\).
Cross - multiply: \(4(3x - 9)=7\times32\).
Expand: \(12x-36 = 224\).
Add 36 to both sides: \(12x=224 + 36=260\).
Divide both sides by 12: \(x=\frac{260}{12}=\frac{65}{3}\approx21.67\).

Answer:

The scale factor is \(\frac{7}{4}\), and \(x = \frac{65}{3}\)