Question

7-19. the polygons below are similar. the figure on the left was reduced to create the polygon on the right. what is the scale factor? find the lengths of the missing sides.

(left trapezoid with top base 5.7 ft, left side 6 ft, bottom base 12 ft, right side x) → (right polygon with left side 4 ft, top side z, right side y, bottom side 6 ft)

hint:
read the math notes box from lesson 7.1.1 if you need a reminder of how to find the scale factor.

Explanation:

Step1: Find Scale Factor

The left polygon has a side of 6 ft, and the corresponding side on the right is 4 ft. Scale factor \( k = \frac{\text{Right side}}{\text{Left side}} = \frac{4}{6} = \frac{2}{3} \).

Step2: Find \( z \)

Corresponding to 5.7 ft on left, \( z = 5.7 \times \frac{2}{3} = 3.8 \) ft.

Step3: Find \( y \)

Corresponding to 12 ft on left, \( y = 12 \times \frac{2}{3} = 8 \) ft.

Step4: Find \( x \)

Corresponding to 6 ft on right (wait, no: right side 6 ft corresponds to left side \( x \)? Wait, no: left side 6 ft → right 4 ft; left \( x \) → right 6 ft? Wait, no, recheck. Wait, the left polygon: sides 6, 5.7, 12, \( x \); right: 4, \( z \), \( y \), 6. So 6 (left) ↔ 4 (right), \( x \) (left) ↔ 6 (right). So scale factor \( \frac{4}{6} = \frac{6}{x} \)? Wait, no, earlier mistake. Wait, similar polygons: corresponding sides proportional. Left side 6 ft (vertical) corresponds to right side 4 ft (vertical). Left side 5.7 ft (top) corresponds to right side \( z \) (top). Left side 12 ft (bottom) corresponds to right side \( y \) (bottom). Left side \( x \) (right leg) corresponds to right side 6 ft (right leg). So correct proportion: \( \frac{6}{4} = \frac{5.7}{z} = \frac{12}{y} = \frac{x}{6} \). Wait, scale factor is \( \frac{\text{Left}}{\text{Right}} = \frac{6}{4} = \frac{3}{2} \)? Wait, I had it reversed. Because left is reduced to right, so scale factor from left to right is \( \frac{4}{6} = \frac{2}{3} \), so right to left is \( \frac{3}{2} \). Wait, no: if left is reduced to right, then right side = left side × scale factor (scale factor <1). So left side 6 → right 4: 6 × k = 4 → k = 4/6 = 2/3. Then left side \( x \) → right side 6: \( x \times \frac{2}{3} = 6 \) → \( x = 6 \div \frac{2}{3} = 9 \) ft. Oh, I messed up earlier. Let's correct:

Step1 (Correct): Scale Factor (Left to Right)

Left side: 6 ft, Right side: 4 ft. Scale factor \( k = \frac{\text{Right}}{\text{Left}} = \frac{4}{6} = \frac{2}{3} \).

Step2 (Correct): \( z \) (top)

Left top: 5.7 ft, so \( z = 5.7 \times k = 5.7 \times \frac{2}{3} = 3.8 \) ft (correct).

Step3 (Correct): \( y \) (bottom)

Left bottom: 12 ft, \( y = 12 \times k = 12 \times \frac{2}{3} = 8 \) ft (correct).

Step4 (Correct): \( x \) (right leg)

Right right leg: 6 ft, so left right leg \( x \): \( x \times k = 6 \) → \( x = 6 \div k = 6 \div \frac{2}{3} = 9 \) ft.

Ah, earlier mistake: the right leg of right polygon is 6 ft, corresponding to left leg \( x \). So \( x \times \frac{2}{3} = 6 \) → \( x = 9 \).

Answer:

Scale factor: \( \frac{2}{3} \)
\( z = 3.8 \) ft, \( y = 8 \) ft, \( x = 9 \) ft