Answer :
To construct triangle \( \Delta PEF \) similar to \( \Delta PQR \) with the given conditions, follow these steps:
1. **Draw \( \Delta PQR \)**:
- Draw a line segment \( PR = 3.5 \, \text{cm} \).
- At point \( P \), draw an angle \( \angle P = 70^\circ \).
- Extend the line segment from \( P \) through \( R \) and mark point \( Q \) such that \( PQ \) is appropriately positioned to form \( \Delta PQR \). Since the exact length of \( PQ \) and \( QR \) are not given, we'll use the similarity condition later to find these.
2. **Determine the lengths for \( \Delta PEF \)**:
- \( PQ = PE = \frac{5}{7} PQ \)
- Given \( PQ \) is divided in the ratio \( 5:7 \), let's call \( PQ = 5k \) and \( PE = 7k \). We'll use this ratio for the construction.
- Since \( \Delta PEF \sim \Delta PQR \), the angles in both triangles are the same and the sides are in proportion.
3. **Construct \( \Delta PEF \) using the given ratio**:
- Draw line segment \( PE \) such that \( PE = 7k \). Since we do not have a specific value for \( k \), you can choose a convenient length for construction.
- Draw \( \angle EPR = 70^\circ \).
- Extend \( PE \) such that it is divided in the ratio \( 5:7 \) from point \( P \). Mark point \( F \) such that \( PF = 5k \).
- Draw a line segment \( EF \) to complete \( \Delta PEF \).
Here’s a step-by-step practical construction approach:
1. **Start with \( PR \)**:
- Draw a line segment \( PR = 3.5 \, \text{cm} \).
2. **Draw \( \angle P = 70^\circ \)**:
- Use a protractor to measure \( \angle P = 70^\circ \) at point \( P \).
3. **Identify the ratio \( 5:7 \)**:
- Assume a convenient length for \( k \) (e.g., 1 cm).
- Draw \( PE \) such that \( PE = 7k \) (e.g., 7 cm if \( k = 1 \, \text{cm} \)).
4. **Construct \( PF \)**:
- Place the compass at \( P \) with a radius equal to \( 5k \) (e.g., 5 cm if \( k = 1 \, \text{cm} \)). Draw an arc intersecting the previously drawn line to mark \( F \).
5. **Complete the triangle \( \Delta PEF \)**:
- Connect \( E \) and \( F \) with a straight line.
This will give you \( \Delta PEF \) similar to \( \Delta PQR \) with \( PE \) in the ratio \( 5:7 \) to \( PQ \).
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