Construct Triangle Given Perimeter and Base Angles
In this construction, we build a triangle when the perimeter (sum of all three sides) and the two base angles are given. The actual side lengths are not known individually.
This is one of the standard constructions in Class 9 NCERT Geometry. The method uses angle bisectors and properties of isosceles triangles to locate the third vertex.
What is Construct Triangle Given Perimeter and Base Angles?
Construct Triangle Given Perimeter and Base Angles Formula
Construction Steps:
- Draw a line segment XY = p (the perimeter).
- At X, construct angle ∠LXY = ∠B.
- At Y, construct angle ∠MYX = ∠C.
- Bisect ∠LXY and ∠MYX. Let these bisectors meet at point A.
- Draw the perpendicular bisector of AX. Let it meet XY at B.
- Draw the perpendicular bisector of AY. Let it meet XY at C.
- Join AB and AC.
- Triangle ABC is the required triangle.
Derivation and Proof
Why this construction works:
- XY = p = perimeter = AB + BC + CA.
- Since B lies on the perpendicular bisector of AX, we have BX = BA.
- Similarly, since C lies on the perpendicular bisector of AY, we have CY = CA.
- Therefore: XB + BC + CY = BA + BC + CA = perimeter = XY ✓
- In △ABX, BX = BA (isosceles), so ∠BAX = ∠BXA.
- ∠ABX = 180° − 2∠BXA. But ∠ABC = ∠ABX + ∠XBA = ... This gives ∠ABC = ∠B as required.
- Similarly, ∠ACB = ∠C as required.
Solved Examples
Example 1: Example 1: Standard construction
Problem: Construct a triangle with perimeter 12 cm, ∠B = 60°, and ∠C = 45°.
Solution:
- Draw XY = 12 cm.
- At X, draw ∠LXY = 60°.
- At Y, draw ∠MYX = 45°.
- Bisect both angles. Let bisectors meet at A.
- Draw perpendicular bisector of AX → meets XY at B.
- Draw perpendicular bisector of AY → meets XY at C.
- Join AB and AC.
Result: △ABC with perimeter 12 cm and ∠B = 60°, ∠C = 45°.
Example 2: Example 2: Another construction
Problem: Construct △PQR with perimeter 14.5 cm, ∠Q = 75°, ∠R = 50°.
Solution:
- Draw XY = 14.5 cm.
- At X, construct 75°. At Y, construct 50°.
- Bisect both angles. Let bisectors meet at P.
- Perpendicular bisectors of PX and PY give Q and R on XY.
- Join PQ and PR.
Verification: Measure PQ + QR + RP — should be 14.5 cm. Measure ∠Q ≈ 75°, ∠R ≈ 50°.
Example 3: Example 3: Equilateral triangle
Problem: Construct an equilateral triangle with perimeter 15 cm using this method.
Solution:
∠B = ∠C = 60°.
- Draw XY = 15 cm.
- At X and Y, construct 60° angles.
- Bisect both (30° each). Bisectors meet at A.
- Perpendicular bisectors of AX and AY give B and C.
Result: Each side = 5 cm. Equilateral triangle constructed.
Example 4: Example 4: Verification problem
Problem: After constructing a triangle with perimeter 11 cm and base angles 50° and 70°, a student measures the sides as 3.2, 3.8, and 4.0 cm. Verify.
Solution:
- Sum = 3.2 + 3.8 + 4.0 = 11.0 cm ✓
- Third angle = 180 − 50 − 70 = 60° (should be verified with protractor)
The construction is correct if perimeter = 11 cm and angles match.
Example 5: Example 5: Finding the third angle
Problem: A triangle has perimeter 10 cm, ∠B = 65°, ∠C = 55°. What is ∠A?
Solution:
- ∠A = 180° − 65° − 55° = 60°
Answer: ∠A = 60°.
Real-World Applications
Applications:
- Engineering: Constructing triangular structures with specific angle and perimeter constraints.
- Surveying: Triangulation when perimeter and angles are known from measurements.
- Design: Creating triangular frames, trusses, and supports.
Key Points to Remember
- This construction works when perimeter and two base angles are given.
- The key idea: use angle bisectors to find the apex vertex, then perpendicular bisectors to locate the base vertices.
- The perpendicular bisector of AX ensures BX = BA (isosceles triangle property).
- XB + BC + CY = BA + BC + CA = perimeter = XY.
- The third angle is automatically determined: ∠A = 180° − ∠B − ∠C.
- This construction is NOT possible if ∠B + ∠C ≥ 180° (no valid triangle).
- Always verify by measuring: check perimeter sum and angle values.
Practice Problems
- Construct a triangle with perimeter 13 cm, ∠B = 50°, ∠C = 60°.
- Construct a triangle with perimeter 10 cm, ∠B = 45°, ∠C = 90°.
- Can you construct a triangle with perimeter 8 cm, ∠B = 100°, ∠C = 90°? Explain.
- Construct an isosceles triangle with perimeter 16 cm and base angles 70° each.
- After construction, measure all three sides and verify that their sum equals the given perimeter.
Frequently Asked Questions
Q1. What information is needed for this construction?
The perimeter (sum of all three sides) and the two base angles. The third angle is automatically determined.
Q2. Why do we draw the perpendicular bisectors?
The perpendicular bisector of AX ensures that point B is equidistant from A and X, making BX = BA. This ensures the perimeter condition is satisfied.
Q3. Can this construction fail?
Yes. If ∠B + ∠C ≥ 180°, no triangle exists. Also, if the angle bisectors don't intersect within the line segment, the construction fails.
Q4. How do you verify the construction?
Measure all three sides and check their sum equals the given perimeter. Measure the base angles with a protractor and check they match.
Q5. What tools are needed?
Ruler, compass, and protractor. The compass is used for angle bisectors and perpendicular bisectors.
Q6. Is this construction exact?
In theory, yes. In practice, small errors in drawing arcs and lines may cause slight deviations.
