Introduction
Triangle construction is a fundamental skill in geometry that involves creating precise triangles using specific given conditions. In this lesson, we'll learn how to construct various types of triangles using only a compass and ruler.
We'll cover different triangle types including equilateral, isosceles, scalene, acute-angled, obtuse-angled, and right-angled triangles under various given conditions.
Essential Tools:
- Compass (for drawing arcs and circles)
- Ruler (for measuring and drawing straight lines)
- Protractor (for measuring angles when needed)
- Pencil (for making construction marks)
Key Concepts:
- An equilateral triangle has all sides equal and all angles equal to 60°
- An isosceles triangle has two equal sides and two equal angles
- A scalene triangle has all sides and angles different
- Triangles can also be classified by their angles: acute, right, or obtuse
Constructing Equilateral Triangle (Given Side)
Method
When one side length is given, follow these steps:
Step 1:
Draw the given side AB using a ruler
Step 2:
With A as center and AB as radius, draw an arc above AB
Step 3:
With B as center and BA as radius, draw another arc intersecting the first arc at C
Step 4:
Join A to C and B to C to complete the equilateral triangle ABC
Justification:
All sides are equal (AB = BC = CA) because they were drawn with the same radius. All angles are 60° because the sum of angles in a triangle is 180°, divided equally.
Constructing Equilateral Triangle (Given Point)
Method
When starting from a given point A and line AX:
Step 1:
Draw point A and line AX
Step 2:
With A as center, draw an arc intersecting AX at B
Step 3:
With same radius, place compass at B and draw an arc intersecting first arc at C
Step 4:
Join A to C and B to C to complete the equilateral triangle ABC
Justification:
This creates a 60° angle at A. Since all construction arcs used the same radius, all sides are equal, making it equilateral.
Constructing Isosceles Right-Angled Triangle
Method
When the base line is given:
Step 1:
Draw the given base line AB
Step 2:
At point A, construct a perpendicular line using compass or protractor
Step 3:
Measure length AB along the perpendicular to locate point C
Step 4:
Join B to C to complete the isosceles right-angled triangle ABC
Justification:
The triangle has one right angle (90°) at A, with sides AB = AC (isosceles), and the hypotenuse BC completes the right triangle.
Constructing Isosceles Triangle (Given Sides)
Method
When all side lengths are given (two equal sides and a base):
Step 1:
Draw the base line AB using the given base length
Step 2:
With A as center and equal side length as radius, draw an arc above AB
Step 3:
With B as center and same equal side length as radius, draw another arc intersecting the first at C
Step 4:
Join A to C and B to C to complete the isosceles triangle ABC
Justification:
The two equal sides (AC = BC) were constructed with equal radii, while the base AB maintains the given length, satisfying the isosceles condition.
Constructing Isosceles Triangle (Given Angles)
Method
When base angles and base side are known:
Step 1:
Draw the given base line AB
Step 2:
At point A, construct the given base angle using a protractor
Step 3:
At point B, construct the same base angle (since it's isosceles)
Step 4:
Extend the angle lines until they meet at point C
Justification:
The two equal base angles guarantee that the sides opposite them (AC and BC) must be equal, making the triangle isosceles with the given base AB.
Constructing Acute/Obtuse/Right Triangles
Method
When a side and two angles are given:
Step 1:
Draw the given side AB
Step 2:
At point A, construct the first given angle
Step 3:
At point B, construct the second given angle
Step 4:
Extend the angle lines until they meet at point C
Justification:
The sum of the three angles must be 180°. The triangle will be:
- Acute if all angles < 90°
- Right if one angle = 90°
- Obtuse if one angle > 90°
Constructing Triangle (Given All Sides)
Method
When lengths of all three sides are given:
Step 1:
Draw the longest side AB (for stability)
Step 2:
With A as center and length AC as radius, draw an arc above AB
Step 3:
With B as center and length BC as radius, draw another arc intersecting the first at C
Step 4:
Join A to C and B to C to complete triangle ABC
Justification:
This construction ensures all sides have exactly the given lengths. The triangle inequality theorem must be satisfied (sum of any two sides > third side).
Constructing Triangle (Given Two Sides & Angle)
Method
When two side lengths and the included angle are given:
Step 1:
Draw one of the given sides AB
Step 2:
At point A, construct the given angle
Step 3:
Measure the second given side length along the angle line to locate point C
Step 4:
Join B to C to complete triangle ABC
Justification:
The construction ensures the two sides have exactly the given lengths and the included angle between them is precisely the given angle. The third side is determined by these conditions.
Practice Exercise
Question 1
Construct an equilateral triangle with sides 6cm long.
Follow the steps for constructing an equilateral triangle with given side length.
Question 2
Construct an isosceles triangle with base 5cm and equal sides 7cm each.
Use the method for constructing isosceles triangle when all sides are given.
Question 3
Construct a right-angled triangle with sides 6cm and 8cm adjacent to the right angle.
Use the method for constructing triangle when two sides and included angle (90°) are given.
Question 4
Construct a triangle with sides 7cm, 5cm, and 4cm.
Use the method for constructing triangle when all three sides are given.
Question 5
Construct a triangle with sides 6cm and 8cm and included angle 45°.
Use the method for constructing triangle when two sides and included angle are given.
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Timed Test in 7 Levels
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