Question

lesson 5 practice problems
a. what is the scale factor from s to l?
b. what is the scale factor from l to s?
c. triangle m is also a scaled copy of s. the scale factor from s to m is $\frac{3}{2}$. what is the scale factor from m to s?

1. are two squares with the same side lengths scaled copies of one another? explain your reasoning.
2. quadrilateral a has side lengths 2, 3, 5, and 6. quadrilateral b has side lengths 4, 5, 8, and 10. could one of the quadrilaterals be a scaled copy of the other? explain.

Explanation:

Step1: Determine if two squares with same side lengths are scaled copies

A scaled - copy is a transformation where all corresponding side lengths are multiplied by a scale factor. If two squares have the same side lengths, the scale factor \(k = \frac{\text{side length of new square}}{\text{side length of original square}}\). Let the side length of the first square be \(s_1\) and of the second square be \(s_2\). Since \(s_1=s_2\), then \(k = \frac{s_2}{s_1}=1\). A scale factor of \(1\) means the two squares are congruent (a special case of a scaled - copy where the figure is unchanged in size).

Step2: Check if quadrilateral B is a scaled - copy of quadrilateral A

For a quadrilateral B to be a scaled - copy of quadrilateral A, the ratios of all corresponding side lengths should be equal.
Let's assume the sides of quadrilateral A are \(a_1 = 2,a_2=3,a_3 = 5,a_4=6\) and sides of quadrilateral B are \(b_1 = 4,b_2 = 5,b_3=8,b_4 = 10\).
Calculate the ratios:

• \(\frac{b_1}{a_1}=\frac{4}{2}=2\)
• \(\frac{b_2}{a_2}=\frac{5}{3}\approx1.67\)
• \(\frac{b_3}{a_3}=\frac{8}{5} = 1.6\)
• \(\frac{b_4}{a_4}=\frac{10}{6}=\frac{5}{3}\approx1.67\)

Since \(\frac{4}{2}
eq\frac{5}{3}
eq\frac{8}{5}
eq\frac{10}{6}\), the ratios of the side lengths are not equal.

Answer:

1. Yes. A scale factor of \(1\) (when the ratio of corresponding side lengths is \(1\)) means they are congruent, and congruent figures are a special case of scaled copies.
2. No. The ratios of the corresponding side lengths (\(\frac{4}{2},\frac{5}{3},\frac{8}{5},\frac{10}{6}\)) are not equal.