Question

a four - sided figure is resized to create a scaled copy. the lengths of its four sides change as in the table below.

original figure	scaled copy
10	65
12	78

find the constant of proportionality from the original figure to the scaled copy. express your answer as a fraction in reduced terms.

Explanation:

Step1: Recall the formula for constant of proportionality

The constant of proportionality \( k \) between two quantities \( y \) (scaled copy) and \( x \) (original figure) is given by \( k=\frac{y}{x} \), where \( y \) is the length of the scaled copy and \( x \) is the length of the original figure.

Step2: Calculate \( k \) for the first pair

For the first pair, \( x = 6 \) and \( y=39 \). So, \( k=\frac{39}{6} \). Simplify this fraction: \( \frac{39\div3}{6\div3}=\frac{13}{2} \).

Step3: Verify with the second pair

For the second pair, \( x = 10 \) and \( y = 65 \). Calculate \( k=\frac{65}{10}=\frac{13}{2} \) (dividing numerator and denominator by 5).

Step4: Verify with the third pair

For the third pair, \( x = 12 \) and \( y = 78 \). Calculate \( k=\frac{78}{12}=\frac{13}{2} \) (dividing numerator and denominator by 6).

Answer:

\(\frac{13}{2}\)