Question

parallelogram c is a scaled copy of parallelogram b. what scale factor takes parallelogram b to parallelogram c?

Explanation:

Step1: Convert mixed numbers to improper fractions

$$ 12\frac{1}{2}=\frac{12\times2 + 1}{2}=\frac{25}{2}, 8\frac{3}{4}=\frac{8\times4+3}{4}=\frac{35}{4},24\frac{1}{2}=\frac{24\times 2+1}{2}=\frac{49}{2} $$

Step2: Calculate the scale factor using side - length ratio

The scale factor \(k\) can be found by comparing corresponding side lengths. Let's use the longer side lengths.
We know that if the original length is \(L_B = 35\) (from parallelogram \(B\)) and the new length is \(L_C=\frac{49}{2}\) (from parallelogram \(C\)).
The scale factor \(k=\frac{L_C}{L_B}\)
Substitute \(L_B = 35=\frac{35}{1}\) and \(L_C=\frac{49}{2}\) into the formula:
\(k=\frac{\frac{49}{2}}{\frac{35}{1}}=\frac{49}{2}\times\frac{1}{35}\)
Simplify the fraction: \(\frac{49}{2}\times\frac{1}{35}=\frac{49\div7}{2\times(35\div7)}=\frac{7}{10}\)

We can also check using the shorter side lengths. Original shorter side \(s_B=\frac{25}{2}\), new shorter side \(s_C = \frac{35}{4}\)
\(k=\frac{s_C}{s_B}=\frac{\frac{35}{4}}{\frac{25}{2}}=\frac{35}{4}\times\frac{2}{25}=\frac{35\times2}{4\times25}=\frac{70}{100}=\frac{7}{10}\)

Answer:

The scale factor is \(\frac{7}{10}\)