Question

If \(\triangle ABC \sim \triangle DEF\) such that DE = 3 cm, EF = 2 cm, DF = 2.5 cm, BC = 4 cm, then the perimeter of \(\triangle ABC\) is :

• A. 7.5 cm
• B. 15 cm
• C. 11.5 cm
• D. 9.5 cm

Hint:

Notice that side \(BC\) is exactly twice the corresponding side \(EF\) (\(4\text{ cm} = 2 \times 2\text{ cm}\)).
This means the scale factor of similarity is \(2\).
Therefore, the perimeter of \(\triangle ABC\) must be exactly twice the perimeter of \(\triangle DEF\):
\[ 2 \times (3 + 2 + 2.5) = 2 \times 7.5 = 15\text{ cm} \]

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from Geometry, specifically focusing on the properties of Similar Triangles.
We are given the side lengths of one triangle and one side of a similar triangle. We need to find the perimeter of the second triangle.

Step 2: Key Formula or Approach:
When two triangles are similar, the ratio of their corresponding sides is equal to the ratio of their perimeters:
\[ \frac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of } \triangle DEF} = \frac{\text{Corresponding side of } \triangle ABC}{\text{Corresponding side of } \triangle DEF} \]
Since \(\triangle ABC \sim \triangle DEF\), the corresponding side of \(BC\) is \(EF\).
Thus, the ratio is \(\frac{BC}{EF}\).

Step 3: Detailed Explanation:
First, calculate the perimeter of \(\triangle DEF\):
\[ \text{Perimeter of } \triangle DEF = DE + EF + DF \]
Substitute the given values:
\[ \text{Perimeter of } \triangle DEF = 3 + 2 + 2.5 = 7.5\text{ cm} \]
Now, identify the corresponding sides:
The side corresponding to \(BC\) in \(\triangle ABC\) is \(EF\) in \(\triangle DEF\).
We are given:
\[ BC = 4\text{ cm} \]
\[ EF = 2\text{ cm} \]
Calculate the ratio of these corresponding sides:
\[ \text{Ratio} = \frac{BC}{EF} = \frac{4}{2} = 2 \]
Now, apply the similarity-perimeter relation:
\[ \frac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of } \triangle DEF} = 2 \]
\[ \text{Perimeter of } \triangle ABC = 2 \times \text{Perimeter of } \triangle DEF \]
Substitute the perimeter of \(\triangle DEF\):
\[ \text{Perimeter of } \triangle ABC = 2 \times 7.5 = 15\text{ cm} \]

Step 4: Final Answer:
The perimeter of \(\triangle ABC\) is \(15\text{ cm}\).