Constructions - Free Printable

Problem Analysis:

The task involves constructing triangles using geometric tools such as a protractor, ruler, and compass. Each part of the problem specifies different sets of given information (side lengths and/or angles) for constructing the triangles. We will solve each part step by step.

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Part 1: Using a Protractor and Ruler

#### a. Construct \( \triangle ABC \) with \( AB = 8 \, \text{cm} \), \( \angle ABC = 40^\circ \), and \( \angle BAC = 54^\circ \).

Steps:
1. Draw side \( AB \):
- Use a ruler to draw a line segment \( AB \) of length 8 cm.

2. Construct \( \angle ABC = 40^\circ \):
- Place the protractor at point \( B \).
- Align the baseline of the protractor with \( AB \).
- Mark a point at \( 40^\circ \) from \( AB \) and draw a ray from \( B \) through this point.

3. Construct \( \angle BAC = 54^\circ \):
- Place the protractor at point \( A \).
- Align the baseline of the protractor with \( AB \).
- Mark a point at \( 54^\circ \) from \( AB \) and draw a ray from \( A \) through this point.

4. Find point \( C \):
- The intersection of the two rays drawn in steps 2 and 3 is point \( C \).

5. Complete the triangle:
- Connect points \( A \) and \( C \), and points \( B \) and \( C \) to form \( \triangle ABC \).

#### b. Construct \( \triangle PQR \) with \( PQ = 6 \, \text{cm} \), \( \angle PQR = 48^\circ \), and \( \angle QPR = 47^\circ \).

Steps:
1. Draw side \( PQ \):
- Use a ruler to draw a line segment \( PQ \) of length 6 cm.

2. Construct \( \angle PQR = 48^\circ \):
- Place the protractor at point \( Q \).
- Align the baseline of the protractor with \( PQ \).
- Mark a point at \( 48^\circ \) from \( PQ \) and draw a ray from \( Q \) through this point.

3. Construct \( \angle QPR = 47^\circ \):
- Place the protractor at point \( P \).
- Align the baseline of the protractor with \( PQ \).
- Mark a point at \( 47^\circ \) from \( PQ \) and draw a ray from \( P \) through this point.

4. Find point \( R \):
- The intersection of the two rays drawn in steps 2 and 3 is point \( R \).

5. Complete the triangle:
- Connect points \( P \) and \( R \), and points \( Q \) and \( R \) to form \( \triangle PQR \).

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Part 2: Using a Ruler and Compass

#### a. Construct \( \triangle ABC \) with \( AB = 10 \, \text{cm} \), \( BC = 8 \, \text{cm} \), and \( AC = 6 \, \text{cm} \).

Steps:
1. Draw side \( AB \):
- Use a ruler to draw a line segment \( AB \) of length 10 cm.

2. Construct arc for \( AC = 6 \, \text{cm} \):
- Place the compass at point \( A \).
- Set the compass radius to 6 cm.
- Draw an arc above \( AB \).

3. Construct arc for \( BC = 8 \, \text{cm} \):
- Place the compass at point \( B \).
- Set the compass radius to 8 cm.
- Draw an arc above \( AB \) that intersects the previous arc.

4. Find point \( C \):
- The intersection of the two arcs is point \( C \).

5. Complete the triangle:
- Connect points \( A \) and \( C \), and points \( B \) and \( C \) to form \( \triangle ABC \).

#### b. Construct \( \triangle PQR \) with \( PQ = 5.6 \, \text{cm} \), \( QR = 4.8 \, \text{cm} \), and \( PR = 4.8 \, \text{cm} \).

Steps:
1. Draw side \( PQ \):
- Use a ruler to draw a line segment \( PQ \) of length 5.6 cm.

2. Construct arc for \( PR = 4.8 \, \text{cm} \):
- Place the compass at point \( P \).
- Set the compass radius to 4.8 cm.
- Draw an arc above \( PQ \).

3. Construct arc for \( QR = 4.8 \, \text{cm} \):
- Place the compass at point \( Q \).
- Set the compass radius to 4.8 cm.
- Draw an arc above \( PQ \) that intersects the previous arc.

4. Find point \( R \):
- The intersection of the two arcs is point \( R \).

5. Complete the triangle:
- Connect points \( P \) and \( R \), and points \( Q \) and \( R \) to form \( \triangle PQR \).

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Part 3: Using a Protractor, Ruler, and Compass

#### a. Construct \( \triangle ABC \) with \( AB = 6 \, \text{cm} \), \( \angle BAC = 85^\circ \), and \( BC = 7 \, \text{cm} \).

Steps:
1. Draw side \( AB \):
- Use a ruler to draw a line segment \( AB \) of length 6 cm.

2. Construct \( \angle BAC = 85^\circ \):
- Place the protractor at point \( A \).
- Align the baseline of the protractor with \( AB \).
- Mark a point at \( 85^\circ \) from \( AB \) and draw a ray from \( A \) through this point.

3. Construct arc for \( BC = 7 \, \text{cm} \):
- Place the compass at point \( B \).
- Set the compass radius to 7 cm.
- Draw an arc that intersects the ray from step 2.

4. Find point \( C \):
- The intersection of the arc and the ray is point \( C \).

5. Complete the triangle:
- Connect points \( A \) and \( C \), and points \( B \) and \( C \) to form \( \triangle ABC \).

#### b. Construct \( \triangle PQR \) with \( PQ = 8 \, \text{cm} \), \( \angle QPR = 78^\circ \), and \( QR = 7 \, \text{cm} \).

Steps:
1. Draw side \( PQ \):
- Use a ruler to draw a line segment \( PQ \) of length 8 cm.

2. Construct \( \angle QPR = 78^\circ \):
- Place the protractor at point \( P \).
- Align the baseline of the protractor with \( PQ \).
- Mark a point at \( 78^\circ \) from \( PQ \) and draw a ray from \( P \) through this point.

3. Construct arc for \( QR = 7 \, \text{cm} \):
- Place the compass at point \( Q \).
- Set the compass radius to 7 cm.
- Draw an arc that intersects the ray from step 2.

4. Find point \( R \):
- The intersection of the arc and the ray is point \( R \).

5. Complete the triangle:
- Connect points \( P \) and \( R \), and points \( Q \) and \( R \) to form \( \triangle PQR \).

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Part 4: Using a Protractor, Ruler, and Compass

#### a. Construct \( \triangle ABC \) with \( AB = 7 \, \text{cm} \), \( AC = 5 \, \text{cm} \), and \( \angle BAC = 55^\circ \).

Steps:
1. Draw side \( AB \):
- Use a ruler to draw a line segment \( AB \) of length 7 cm.

2. Construct \( \angle BAC = 55^\circ \):
- Place the protractor at point \( A \).
- Align the baseline of the protractor with \( AB \).
- Mark a point at \( 55^\circ \) from \( AB \) and draw a ray from \( A \) through this point.

3. Construct arc for \( AC = 5 \, \text{cm} \):
- Place the compass at point \( A \).
- Set the compass radius to 5 cm.
- Draw an arc that intersects the ray from step 2.

4. Find point \( C \):
- The intersection of the arc and the ray is point \( C \).

5. Complete the triangle:
- Connect points \( A \) and \( C \), and points \( B \) and \( C \) to form \( \triangle ABC \).

#### b. Construct \( \triangle PQR \) with \( PQ = 7.5 \, \text{cm} \), \( PR = 6.8 \, \text{cm} \), and \( \angle QPR = 75^\circ \).

Steps:
1. Draw side \( PQ \):
- Use a ruler to draw a line segment \( PQ \) of length 7.5 cm.

2. Construct \( \angle QPR = 75^\circ \):
- Place the protractor at point \( P \).
- Align the baseline of the protractor with \( PQ \).
- Mark a point at \( 75^\circ \) from \( PQ \) and draw a ray from \( P \) through this point.

3. Construct arc for \( PR = 6.8 \, \text{cm} \):
- Place the compass at point \( P \).
- Set the compass radius to 6.8 cm.
- Draw an arc that intersects the ray from step 2.

4. Find point \( R \):
- The intersection of the arc and the ray is point \( R \).

5. Complete the triangle:
- Connect points \( P \) and \( R \), and points \( Q \) and \( R \) to form \( \triangle PQR \).

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Final Answer:

\[
\boxed{\text{See detailed steps above for each construction.}}
\]