Given \(\triangle ABC \sim \triangle EDF\) with \(AB=5\,\text{cm}\), \(AC=7\,\text{cm}\), \(DE=12\,\text{cm}\), \(DF=15\,\text{cm}\). Find the remaining sides.
\(BC=\dfrac{25}{4}\,\text{cm}=6.25\,\text{cm},\; EF=\dfrac{84}{5}\,\text{cm}=16.8\,\text{cm}.\)
Step 1: The triangles are similar: \(\triangle ABC \sim \triangle EDF\).
This means their corresponding sides are in the same ratio. The order of letters tells us the matching parts:
Step 2: Write the proportion for the sides:
\[ \dfrac{AB}{DE} = \dfrac{AC}{EF} = \dfrac{BC}{DF} \]
Step 3: First, find \(EF\).
\[ \dfrac{AB}{DE} = \dfrac{AC}{EF} \]
Substitute the values: \( \dfrac{5}{12} = \dfrac{7}{EF} \).
Cross multiply: \( 5 \times EF = 12 \times 7 = 84 \).
So, \( EF = \dfrac{84}{5} = 16.8\,\text{cm} \).
Step 4: Now, find \(BC\).
\[ \dfrac{AB}{DE} = \dfrac{BC}{DF} \]
Substitute the values: \( \dfrac{5}{12} = \dfrac{BC}{15} \).
Cross multiply: \( 5 \times 15 = 12 \times BC \).
So, \( 75 = 12 \times BC \).
\( BC = \dfrac{75}{12} = \dfrac{25}{4} = 6.25\,\text{cm} \).
Final Answer:
\(EF = 16.8\,\text{cm},\; BC = 6.25\,\text{cm}.\)