Question

guided practice
see example 1 ( \triangle xyz sim \triangle pqr ) in each pair. find the unknown lengths.
1.
2.
see example 2 3. the rectangular gardens at right are similar in shape. how wide is the smaller garden?
see example 3 4. a water tower casts a shadow that is 21 ft long. a tree casts a shadow that is 8 ft long. estimate the height of the water tower.
independent practice
see example 1 ( \triangle abc sim \triangle def ) in each pair. find the unknown lengths.
5.
6.
see example 2 7. the two rectangular windows at right are similar. what is the height of the bigger window?
see example 3 8. a cactus casts a shadow that is 14 ft 7 in. long. a gate nearby casts a shadow that is 5 ft long. estimate the height of the cactus.

Explanation:

Step1: Find similarity ratio (Q1)

$\text{Ratio} = \frac{20}{12} = \frac{5}{3}$

Step2: Solve for $a$ (Q1)

$a = 9 \times \frac{5}{3} = 15$ cm

Step3: Solve for $y$ (Q1)

$y = 30 \times \frac{3}{5} = 18$ cm

Step4: Find similarity ratio (Q2)

$\text{Ratio} = \frac{30}{48} = \frac{5}{8}$

Step5: Solve for $y$ (Q2)

$y = 35 \times \frac{8}{5} = 56$ m

Step6: Solve for unknown side (Q2)

$\text{Unknown side} = 64 \times \frac{5}{8} = 40$ m

Step7: Set up proportion (Q3)

$\frac{\text{Small width}}{36} = \frac{42}{54}$

Step8: Solve small width (Q3)

$\text{Small width} = 36 \times \frac{42}{54} = 28$ ft

Step9: Set up proportion (Q4)

$\frac{x}{9.5} = \frac{21}{8}$

Step10: Solve tower height (Q4)

$x = 9.5 \times \frac{21}{8} = 24.9375$ ft

Step11: Find similarity ratio (Q5)

$\text{Ratio} = \frac{21}{14} = \frac{3}{2}$

Step12: Solve for $x$ (Q5)

$x = 12 \times \frac{3}{2} = 18$ in

Step13: Find similarity ratio (Q6)

$\text{Ratio} = \frac{4}{7.2} = \frac{5}{9}$

Step14: Solve for $b$ (Q6)

$b = 12.96 \times \frac{5}{9} = 7.2$ ft

Step15: Set up proportion (Q7)

$\frac{\text{Big height}}{3} = \frac{5.2}{4}$

Step16: Solve big height (Q7)

$\text{Big height} = 3 \times \frac{5.2}{4} = 3.9$ ft

Step17: Convert units (Q8)

$3$ ft $4$ in $= 40$ in, $14$ ft $7$ in $= 175$ in, $5$ ft $= 60$ in

Step18: Set up proportion (Q8)

$\frac{\text{Cactus height}}{40} = \frac{175}{60}$

Step19: Solve cactus height (Q8)

$\text{Cactus height} = 40 \times \frac{175}{60} \approx 116.67$ in $= 9$ ft $8.67$ in

Answer:

1. $a = 15$ cm, $y = 18$ cm
2. $y = 56$ m, unknown side $= 40$ m
3. 28 ft
4. 24.94 ft (rounded)
5. $x = 18$ in
6. $b = 7.2$ ft
7. 3.9 ft
8. ~9 ft 9 in (or 116.67 in)