Question

a square pyramid has a base side length of 20.4 cm and a height of 18.2 cm. which square pyramid is similar?
base side length = 18.2 cm, height = 20.4 cm
base side length = 22.9 cm, height = 20.7 cm
base side length = 30.6 cm, height = 27.3 cm
base side length = 40.8 cm, height = 45.5 cm

Explanation:

Step1: Recall similarity of solids

For two square pyramids to be similar, the ratio of their corresponding linear dimensions (base side length and height) must be equal. So we need to find the ratio of base side length to height for the given pyramid and each option, then check which ratio matches.

Step2: Calculate ratio for given pyramid

Given base side length \( b = 20.4 \, \text{cm} \) and height \( h = 18.2 \, \text{cm} \). The ratio \( \frac{b}{h}=\frac{20.4}{18.2}=\frac{204}{182}=\frac{102}{91}\approx1.1209 \)

Step3: Calculate ratio for Option 1

Base side length \( = 18.2 \, \text{cm} \), height \( = 20.4 \, \text{cm} \). Ratio \( \frac{18.2}{20.4}=\frac{182}{204}=\frac{91}{102}\approx0.8922 \). Not equal to \( \frac{102}{91} \)

Step4: Calculate ratio for Option 2

Base side length \( = 22.9 \, \text{cm} \), height \( = 20.7 \, \text{cm} \). Ratio \( \frac{22.9}{20.7}\approx1.1063 \). Not equal to \( \frac{102}{91}\approx1.1209 \)

Step5: Calculate ratio for Option 3

Base side length \( = 30.6 \, \text{cm} \), height \( = 27.3 \, \text{cm} \). Ratio \( \frac{30.6}{27.3}=\frac{306}{273}=\frac{102}{91}\approx1.1209 \). This matches the ratio of the given pyramid.

Step6: Calculate ratio for Option 4 (for completeness)

Base side length \( = 40.8 \, \text{cm} \), height \( = 45.5 \, \text{cm} \). Ratio \( \frac{40.8}{45.5}=\frac{408}{455}=\frac{24}{26.7647}\approx0.8968 \). Not equal to \( \frac{102}{91} \)

Answer:

C. base side length = 30.6 cm, height = 27.3 cm (Note: Using the option numbering as per the order, the third option is the correct one with base side length 30.6 cm and height 27.3 cm)