Question

problem 3
the pentagons are scaled copies of one another.
a) determine the scale factor.
scale factor:_______
b) use the scale factor to find x.
x =_______

problem 4
draw a scaled copy of the polygon with scale factor 3.
(hint: label the parts of the figures)

Explanation:

Part (a) - Determine the Scale Factor

Step 1: Identify Corresponding Sides

The two pentagons are scaled copies, so we can use the corresponding vertical sides (9.6 cm and 12 cm) to find the scale factor. Let the scale factor from the smaller to the larger pentagon be \( k \). The formula for scale factor is \( \text{Scale Factor} = \frac{\text{Length of side in larger figure}}{\text{Length of corresponding side in smaller figure}} \).

Step 2: Calculate the Scale Factor

Using the vertical sides: \( k = \frac{12}{9.6} \)
Simplify \( \frac{12}{9.6} \): Multiply numerator and denominator by 10 to eliminate decimals: \( \frac{120}{96} \). Divide numerator and denominator by 24: \( \frac{120\div24}{96\div24} = \frac{5}{4} = 1.25 \)

Step 1: Recall the Scale Factor Relationship

For scaled copies, the ratio of corresponding sides is equal to the scale factor. The side of length 8.4 cm in the smaller pentagon corresponds to side \( x \) in the larger pentagon. So, \( \frac{x}{8.4} = \text{Scale Factor} \) (which we found as \( \frac{5}{4} \)).

Step 2: Solve for \( x \)

Using the equation \( x = 8.4 \times \text{Scale Factor} \). Substitute the scale factor \( \frac{5}{4} \):
\( x = 8.4 \times \frac{5}{4} \)
First, calculate \( 8.4 \div 4 = 2.1 \), then multiply by 5: \( 2.1 \times 5 = 10.5 \)

Answer:

(a):
Scale Factor: \( \frac{5}{4} \) (or 1.25)

Part (b) - Use the Scale Factor to Find \( x \)