Worksheet on Construction of Triangles - Draw accurate triangles using the given side lengths.

Problem Description:

The task is to construct triangles using the given side lengths. This involves drawing accurate triangles based on the provided dimensions for each triangle. The method used is the side-side-side (SSS) construction, where all three sides of the triangle are known.

Solution Approach:

To construct a triangle with given side lengths, follow these steps:

1. Verify the Triangle Inequality Theorem: Before constructing the triangle, ensure that the given side lengths satisfy the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is not met, a triangle cannot be constructed.

2. Use a Ruler and Compass:
- Draw one side of the triangle as a base.
- Use a compass to draw arcs from the endpoints of the base with radii equal to the other two side lengths.
- The intersection of these arcs will give the third vertex of the triangle.
- Connect the third vertex to the endpoints of the base to complete the triangle.

3. Repeat for Each Triangle: Apply the above steps to construct each triangle listed in the worksheet.

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Step-by-Step Construction for Each Triangle:

#### Triangle 1:
- Sides: 7 cm, 5 cm, 9 cm
- Verification:
- \(7 + 5 > 9\) (True)
- \(7 + 9 > 5\) (True)
- \(5 + 9 > 7\) (True)
- All conditions are satisfied.
- Construction:
1. Draw a line segment \(AB = 9 \, \text{cm}\).
2. With \(A\) as the center, draw an arc of radius \(7 \, \text{cm}\).
3. With \(B\) as the center, draw an arc of radius \(5 \, \text{cm}\).
4. Mark the intersection of the arcs as point \(C\).
5. Join \(AC\) and \(BC\) to complete the triangle.

#### Triangle 2:
- Sides: 2.5 cm, 2.5 cm, 2.5 cm
- Verification:
- \(2.5 + 2.5 > 2.5\) (True)
- \(2.5 + 2.5 > 2.5\) (True)
- \(2.5 + 2.5 > 2.5\) (True)
- All conditions are satisfied.
- Construction:
1. Draw a line segment \(AB = 2.5 \, \text{cm}\).
2. With \(A\) as the center, draw an arc of radius \(2.5 \, \text{cm}\).
3. With \(B\) as the center, draw an arc of radius \(2.5 \, \text{cm}\).
4. Mark the intersection of the arcs as point \(C\).
5. Join \(AC\) and \(BC\) to complete the equilateral triangle.

#### Triangle 3:
- Sides: 4 cm, 7.5 cm, 8 cm
- Verification:
- \(4 + 7.5 > 8\) (True)
- \(4 + 8 > 7.5\) (True)
- \(7.5 + 8 > 4\) (True)
- All conditions are satisfied.
- Construction:
1. Draw a line segment \(AB = 8 \, \text{cm}\).
2. With \(A\) as the center, draw an arc of radius \(4 \, \text{cm}\).
3. With \(B\) as the center, draw an arc of radius \(7.5 \, \text{cm}\).
4. Mark the intersection of the arcs as point \(C\).
5. Join \(AC\) and \(BC\) to complete the triangle.

#### Triangle 4:
- Sides: 3.8 cm, 4.2 cm, 6.2 cm
- Verification:
- \(3.8 + 4.2 > 6.2\) (True)
- \(3.8 + 6.2 > 4.2\) (True)
- \(4.2 + 6.2 > 3.8\) (True)
- All conditions are satisfied.
- Construction:
1. Draw a line segment \(AB = 6.2 \, \text{cm}\).
2. With \(A\) as the center, draw an arc of radius \(3.8 \, \text{cm}\).
3. With \(B\) as the center, draw an arc of radius \(4.2 \, \text{cm}\).
4. Mark the intersection of the arcs as point \(C\).
5. Join \(AC\) and \(BC\) to complete the triangle.

#### Triangle 5:
- Sides: 1.7 cm, 8.2 cm, 7.2 cm
- Verification:
- \(1.7 + 8.2 > 7.2\) (True)
- \(1.7 + 7.2 > 8.2\) (False)
- \(8.2 + 7.2 > 1.7\) (True)
- The second condition is not satisfied.
- Conclusion: A triangle cannot be constructed with these side lengths.

#### Triangle 6:
- Sides: 6.6 cm, 6.6 cm, 6.6 cm
- Verification:
- \(6.6 + 6.6 > 6.6\) (True)
- \(6.6 + 6.6 > 6.6\) (True)
- \(6.6 + 6.6 > 6.6\) (True)
- All conditions are satisfied.
- Construction:
1. Draw a line segment \(AB = 6.6 \, \text{cm}\).
2. With \(A\) as the center, draw an arc of radius \(6.6 \, \text{cm}\).
3. With \(B\) as the center, draw an arc of radius \(6.6 \, \text{cm}\).
4. Mark the intersection of the arcs as point \(C\).
5. Join \(AC\) and \(BC\) to complete the equilateral triangle.

#### Triangle 7:
- Sides: 9.2 cm, 3.5 cm, 8.4 cm
- Verification:
- \(9.2 + 3.5 > 8.4\) (True)
- \(9.2 + 8.4 > 3.5\) (True)
- \(3.5 + 8.4 > 9.2\) (True)
- All conditions are satisfied.
- Construction:
1. Draw a line segment \(AB = 8.4 \, \text{cm}\).
2. With \(A\) as the center, draw an arc of radius \(9.2 \, \text{cm}\).
3. With \(B\) as the center, draw an arc of radius \(3.5 \, \text{cm}\).
4. Mark the intersection of the arcs as point \(C\).
5. Join \(AC\) and \(BC\) to complete the triangle.

#### Triangle 8:
- Sides: 3.7 cm, 3.9 cm, 5.2 cm
- Verification:
- \(3.7 + 3.9 > 5.2\) (True)
- \(3.7 + 5.2 > 3.9\) (True)
- \(3.9 + 5.2 > 3.7\) (True)
- All conditions are satisfied.
- Construction:
1. Draw a line segment \(AB = 5.2 \, \text{cm}\).
2. With \(A\) as the center, draw an arc of radius \(3.7 \, \text{cm}\).
3. With \(B\) as the center, draw an arc of radius \(3.9 \, \text{cm}\).
4. Mark the intersection of the arcs as point \(C\).
5. Join \(AC\) and \(BC\) to complete the triangle.

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Final Answer:

\[
\boxed{\text{Triangles 1, 2, 3, 4, 6, 7, and 8 can be constructed. Triangle 5 cannot be constructed.}}
\]