Properties of Triangles Revision Notes for Leaving Cert Mathematics

Properties of Triangles

Overview

Triangles are essential geometric shapes with unique properties based on their sides, angles, and structural elements. Understanding these properties helps in problem-solving and constructing geometric proofs.

Types of Triangles

By Sides:

Equilateral Triangle: All three sides are equal; all angles are 60°.
Isosceles Triangle: Two sides are equal; the angles opposite these sides are also equal.
Scalene Triangle: All sides and angles are different.

By Angles:

Acute Triangle: All angles are less than 90°.
Obtuse Triangle: One angle is greater than 90°.
Right Triangle: One angle is exactly 90°, with the side opposite it called the hypotenuse.

Key Properties

Angle Sum Property: The sum of the angles in a triangle is always $180^\circ$ .
Exterior Angle Property: An exterior angle is equal to the sum of the two non-adjacent interior angles.
Inequality Property: The sum of the lengths of any two sides of a triangle is greater than the third side (Triangle Inequality Theorem).
Pythagoras' Theorem (Right Triangles): In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Special Points:

Centroid: Intersection of medians, dividing them in a 2:1 ratio.
Orthocenter: Intersection of altitudes.
Circumcenter: Intersection of perpendicular bisectors, the centre of a circle passing through all vertices.
Incenter: Intersection of angle bisectors, the centre of the inscribed circle.

Worked Examples

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Example 1: Angle Sum Property

Problem: In a triangle, two angles measure 50° and 70°

Find the third angle.

Solution:

Step 1: Use the angle sum property:

\text{Sum of angles} = 180^\circ

Step 2: Subtract the known angles:

\text{Third angle} = 180^\circ - (50^\circ + 70^\circ) = 60^\circ

Answer: The third angle is 60°

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Example 2: Triangle Inequality Property

Problem: Can a triangle have sides of lengths 3, 4, and 8?

Solution:

Step 1: Check the inequality property:

a + b > c, \, b + c > a, \, \text{and } a + c > b

Step 2: Substitute the side lengths:

3 + 4 = 7 \not> 8

Conclusion: The sides do not satisfy the triangle inequality property.

Answer: A triangle cannot have sides of lengths 3, 4, and 8.

Summary

Types of Triangles: Categorised by sides (equilateral, isosceles, scalene) and angles (acute, obtuse, right).
Key Properties:
- Angle sum is always 180°
- The exterior angle equals the sum of two non-adjacent interior angles.
- Satisfies the triangle inequality property.
- Special points include the centroid, orthocenter, circumcenter, and incenter.
Applications: Used in constructions, trigonometry, and proofs.

Properties of Triangles (Leaving Cert Mathematics): Revision Notes