Question

quadrilateral t is a scaled copy of quadrilateral s.
quadrilateral s has a side labeled ( 5\frac{1}{2} ) and another labeled ( 3\frac{1}{2} ). quadrilateral t has a side labeled ( 4\frac{2}{5} ) and a side labeled ( r ).
what is the value of ( r )?

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \(5\frac{1}{2}\) to an improper fraction: \(5\frac{1}{2}=\frac{5\times2 + 1}{2}=\frac{11}{2}\).
Then, convert \(4\frac{2}{5}\) to an improper fraction: \(4\frac{2}{5}=\frac{4\times5+2}{5}=\frac{22}{5}\).
Also, convert \(3\frac{1}{2}\) to an improper fraction: \(3\frac{1}{2}=\frac{3\times2 + 1}{2}=\frac{7}{2}\).

Step2: Find the scale factor

Since quadrilateral \(T\) is a scaled copy of \(S\), the ratio of corresponding sides is the scale factor. Let the scale factor be \(k\). So, \(k=\frac{\text{Side of }T}{\text{Corresponding side of }S}=\frac{\frac{22}{5}}{\frac{11}{2}}\).
Simplify the ratio: \(\frac{\frac{22}{5}}{\frac{11}{2}}=\frac{22}{5}\times\frac{2}{11}=\frac{4}{5}\).

Step3: Calculate \(r\)

Now, use the scale factor to find \(r\). The corresponding side of \(r\) in \(S\) is \(3\frac{1}{2}=\frac{7}{2}\). So, \(r=\frac{7}{2}\times\frac{4}{5}\).
Simplify the multiplication: \(\frac{7}{2}\times\frac{4}{5}=\frac{28}{10}=\frac{14}{5}=2\frac{4}{5}\).

Answer:

\(2\frac{4}{5}\) (or \(\frac{14}{5}\))