Constructing Triangles (SSS)
When all three sides of a triangle are given, you can construct it using a compass and ruler. This is the SSS construction.
The key requirement: the three sides must satisfy the triangle inequality — the sum of any two sides must be greater than the third side. If this condition fails, the triangle cannot be constructed.
What is Constructing Triangles (SSS) - Grade 7 Maths (Practical Geometry)?
Steps to construct a triangle given three sides a, b, c:
- Draw the longest side as the base using a ruler.
- With one end of the base as centre, draw an arc of radius equal to the second side.
- With the other end as centre, draw an arc of radius equal to the third side.
- The two arcs intersect at a point — this is the third vertex.
- Join this point to both ends of the base.
Constructing Triangles (SSS) Formula
Triangle Inequality (must check first):
a + b > c, b + c > a, a + c > b
If any one condition fails, the triangle cannot be constructed.
Types and Properties
Cases:
- All sides different (scalene): One unique triangle (up to reflection).
- Two sides equal (isosceles): The two arcs give a point above or below the base.
- All sides equal (equilateral): Both arcs have the same radius as the base.
Solved Examples
Example 1: Scalene Triangle
Problem: Construct △ABC with AB = 5 cm, BC = 6 cm, CA = 7 cm.
Solution:
- Check triangle inequality: 5+6=11>7, 6+7=13>5, 5+7=12>6. ✓
- Draw AB = 5 cm (base).
- From A, draw arc of radius 7 cm (= CA).
- From B, draw arc of radius 6 cm (= BC).
- Arcs intersect at C. Join AC and BC.
Answer: △ABC is constructed with sides 5, 6, 7 cm.
Example 2: Equilateral Triangle
Problem: Construct an equilateral triangle with side 4 cm.
Solution:
- Draw AB = 4 cm.
- From A, draw arc of radius 4 cm.
- From B, draw arc of radius 4 cm.
- Arcs intersect at C. Join AC and BC.
Answer: Equilateral triangle with all sides = 4 cm.
Example 3: Triangle Inequality Failure
Problem: Can you construct a triangle with sides 2, 3, 7?
Solution:
- Check: 2 + 3 = 5 < 7. Triangle inequality fails.
Answer: No, this triangle cannot be constructed.
Real-World Applications
Real-world uses:
- Land surveying: Creating triangular plots from measured side lengths.
- Engineering: Building triangular frames with specific dimensions.
- Architecture: Designing triangular roof trusses.
Key Points to Remember
- SSS construction requires all three sides given.
- Always check the triangle inequality first.
- Use compass to draw arcs and find the third vertex.
- The intersection of two arcs gives the third vertex.
- The constructed triangle is unique (up to reflection).
Practice Problems
- Construct a triangle with sides 4, 5, 6 cm.
- Construct a triangle with sides 3, 3, 3 cm.
- Can a triangle be constructed with sides 1, 2, 4 cm?
- Construct a triangle with sides 5, 5, 8 cm.
Frequently Asked Questions
Q1. What is SSS construction?
Constructing a triangle when all three side lengths are given, using compass and ruler.
Q2. When does SSS construction fail?
When the triangle inequality is violated — when the sum of two sides is not greater than the third side.
Q3. Is the triangle unique?
Yes (up to reflection). Given three fixed side lengths satisfying the triangle inequality, only one triangle shape is possible.
Related Topics
- Constructing Triangles (SAS)
- Constructing Triangles (ASA)
- SSS Congruence Rule
- Triangle Inequality Property
- Constructing a Line Segment
- Constructing Perpendicular Lines
- Constructing Angles
- Bisecting a Line Segment
- Bisecting an Angle
- Constructing Parallel Lines
- Constructing Triangles (RHS)
- Constructing Quadrilaterals
- Constructing Special Quadrilaterals
- Triangle Construction Problems
