Question

parallelogram c is a scaled copy of parallelogram b. parallelogram b has a top side labeled ( 12\frac{1}{2} ) and a right side labeled 35. parallelogram c has a top side labeled ( 8\frac{3}{4} ) and a right side labeled ( 24\frac{1}{2} ). what scale factor takes parallelogram b to parallelogram c?

Explanation:

Step1: Convert mixed numbers to improper fractions

For the length of parallelogram B: \(12\frac{1}{2}=\frac{12\times2 + 1}{2}=\frac{25}{2}\), and its height is \(35=\frac{35}{1}\).
For the length of parallelogram C: \(8\frac{3}{4}=\frac{8\times4+3}{4}=\frac{35}{4}\), and its height is \(24\frac{1}{2}=\frac{24\times2 + 1}{2}=\frac{49}{2}\).
We can use either the length or the height to find the scale factor. Let's use the length first. The scale factor \(k\) is the ratio of the corresponding side of C to the corresponding side of B. So \(k=\frac{\text{Length of C}}{\text{Length of B}}\).

Step2: Calculate the scale factor

Substitute the values: \(k = \frac{\frac{35}{4}}{\frac{25}{2}}\). When dividing fractions, we multiply by the reciprocal: \(k=\frac{35}{4}\times\frac{2}{25}\). Simplify the numerator and denominator: \(35\) and \(25\) have a common factor of \(5\), \(2\) and \(4\) have a common factor of \(2\). So \(\frac{35\div5}{4\div2}\times\frac{2\div2}{25\div5}=\frac{7}{2}\times\frac{1}{5}=\frac{7}{10}\)? Wait, no, let's do it again. \(\frac{35}{4}\times\frac{2}{25}=\frac{35\times2}{4\times25}=\frac{70}{100}=\frac{7}{10}\)? Wait, no, maybe use the height. Let's check the height: \(\frac{\text{Height of C}}{\text{Height of B}}=\frac{\frac{49}{2}}{35}=\frac{49}{2}\times\frac{1}{35}=\frac{49}{70}=\frac{7}{10}\). Wait, but let's check the length again. Wait, \(12\frac{1}{2}\) is \(\frac{25}{2}\), \(8\frac{3}{4}\) is \(\frac{35}{4}\). So \(\frac{35}{4}\div\frac{25}{2}=\frac{35}{4}\times\frac{2}{25}=\frac{70}{100}=\frac{7}{10}\). Wait, but let's check with another pair. Wait, maybe I made a mistake. Wait, parallelogram C is a scaled copy of B, so scale factor from B to C is (C's side)/(B's side). Let's take the height: B's height is 35, C's height is \(24\frac{1}{2}=\frac{49}{2}\). So \(\frac{49}{2}\div35=\frac{49}{2}\times\frac{1}{35}=\frac{49}{70}=\frac{7}{10}\). Or length: B's length is \(12\frac{1}{2}=\frac{25}{2}\), C's length is \(8\frac{3}{4}=\frac{35}{4}\). \(\frac{35}{4}\div\frac{25}{2}=\frac{35}{4}\times\frac{2}{25}=\frac{70}{100}=\frac{7}{10}\). Yes, that's correct. Alternatively, let's convert to decimals. \(12\frac{1}{2}=12.5\), \(8\frac{3}{4}=8.75\). \(8.75\div12.5 = 0.7=\frac{7}{10}\). And \(35\) and \(24.5\): \(24.5\div35 = 0.7=\frac{7}{10}\). So the scale factor is \(\frac{7}{10}\) or \(0.7\).

Answer:

\(\frac{7}{10}\) (or \(0.7\))