Question

a photocopy of a polygon was reduced to 35% of its original size. then, the copy was again reduced to 35% of its size
a. is the second photocopy similar to the original?
b. what is the ratio of the corresponding sides of the second photocopy to the original?
a. is the second photocopy similar to the original?
○ a. yes, because the angles stay the same but the corresponding sides are not proportional
○ b. yes, because the angles stay the same and the corresponding sides are proportional
○ c. no, because the angles are not the same and the corresponding sides are proportional
○ d. no, because the angles stay the same and the corresponding sides are proportional

Explanation:

a.

Step1: Recall the definition of similar polygons

Similar polygons have equal corresponding angles and proportional corresponding sides. When a polygon is scaled (reduced or enlarged), the angles remain the same (since it's a rigid - transformation in terms of angle - measure) and the sides are scaled by a factor. Here, each reduction is a uniform scaling.

Step2: Analyze the angle and side - length properties

The first reduction to 35% of the original size and the second reduction to 35% of the first - reduced size are both uniform scalings. The angles of the polygon do not change during scaling (a property of similarity transformations like dilation). The corresponding sides of the second photocopy to the first photocopy are in proportion (scale factor 0.35), and the corresponding sides of the first photocopy to the original are in proportion (scale factor 0.35). By the transitive property of proportion (\(\frac{a}{b}=\frac{b}{c}\Rightarrow\frac{a}{c}\)), the corresponding sides of the second photocopy to the original are in proportion.

b.

Step1: Calculate the combined scale factor

Let the scale factor of the first reduction be \(k_1 = 0.35\) and the scale factor of the second reduction be \(k_2=0.35\). The scale factor \(k\) of the second photocopy with respect to the original is the product of the two scale factors. Using the formula \(k = k_1\times k_2\).

Step2: Compute the value of \(k\)

Substitute \(k_1 = 0.35\) and \(k_2 = 0.35\) into the formula \(k=k_1\times k_2\). We get \(k=(0.35)\times(0.35)= 0.1225=\frac{1225}{10000}=\frac{49}{400}\)

Answer:

a. B. Yes, because the angles stay the same and the corresponding sides are proportional
b. The ratio of the corresponding sides of the second photocopy to the original is \(\frac{49}{400}\) (or \(0.1225\))