In `Delta ABC and Delta MNP` , if AB=2.25 cm, MP = 4.5 cm and PN = 7.5 cm , ` angle` ACB =` angle MNP` and `angle ABC` = `angle MPN`, then what is the length (in cm) of side BC?

Options

A. `3.75`
B. `4.75`
C. `3.5`
D. `4.5`

Answer

To solve the problem, we will use the concept of similar triangles. Given that triangles ABC and MNP are similar, we can set up a proportion based on their corresponding sides.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- In triangle ABC:
- \( AB = 2.25 \, \text{cm} \)
- Let \( BC = x \) (the side we need to find)
- In triangle MNP:
- \( MP = 4.5 \, \text{cm} \)
- \( PN = 7.5 \, \text{cm} \)
- Angles:
- \( \angle ACB = \angle MNP \)
- \( \angle ABC = \angle MPN \)

2. **Establish Similarity:**
- Since two angles of triangle ABC are equal to two angles of triangle MNP, by the Angle-Angle (AA) criterion, triangle ABC is similar to triangle MNP.
- Therefore, we can write:
\[ \frac{MP}{AB} = \frac{PN}{BC} \]

3. **Substitute the Known Values:**
- Substitute the known lengths into the proportion:
\[ \frac{4.5}{2.25} = \frac{7.5}{x} \]

4. **Cross-Multiply to Solve for x:**
- Cross-multiplying gives:
\[ 4.5 \cdot x = 2.25 \cdot 7.5 \]

5. **Calculate the Right Side:**
- Calculate \( 2.25 \cdot 7.5 \):
\[ 2.25 \cdot 7.5 = 16.875 \]
- Now we have:
\[ 4.5x = 16.875 \]

6. **Isolate x:**
- Divide both sides by 4.5:
\[ x = \frac{16.875}{4.5} \]

7. **Calculate x:**
- Performing the division:
\[ x = 3.75 \]

### Final Answer:
The length of side \( BC \) is \( 3.75 \, \text{cm} \).

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