Triangles: Practice Questions & Notes

A triangle is a closed 2D figure formed by joining three non-collinear points with three line segments. It has:

3 sides
3 angles
3 vertices

Think of it as the simplest polygon — and yet, foundational in almost all of geometry.

Classification of Triangles

Based on Sides:

Type	Description
Scalene	All sides and all angles are unequal
Isosceles	Two sides (and their opposite angles) are equal
Equilateral	All sides and angles are equal (each angle = 60°)

Based on Angles:

Type	Description
Acute-angled	All angles are less than 90°
Right-angled	One angle is exactly 90°
Obtuse-angled	One angle is greater than 90°

Angle Sum Property

Sum of the interior angles of a triangle is always:

\angle A + \angle B + \angle C = 180^\circ

This is true for every triangle, regardless of its type.

Exterior Angle Property

The exterior angle of a triangle equals the sum of the two opposite interior angles.

\text{Exterior angle} = \text{Interior 1} + \text{Interior 2}

This is a crucial property used in solving many MCQs quickly.

Triangle Inequality Theorem

For any triangle:

\text{Sum of any two sides} > \text{Third side}

This gives us three important inequalities:

$AB + BC > AC$
$AB + AC > BC$
$AC + BC > AB$

Also:

Difference of any two sides < third side

Special Triangles and Properties

1. Equilateral Triangle

All sides equal, all angles = 60°
Area:

\text{Area} = \frac{\sqrt{3}}{4}a^2

2. Isosceles Triangle

Two sides equal
Base angles are equal
Height from apex bisects base and vertex angle

3. Right-Angled Triangle

One angle = 90°
Pythagoras Theorem:
If $a^2 + b^2 = c^2$ , triangle is right-angled (where $c$ is hypotenuse)

Important Formulas

1. Area of a Triangle

Standard formula (using base and height):

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

Heron’s Formula (when all sides are known):

s = \frac{a + b + c}{2} \quad \text{(semi-perimeter)} \\ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Using two sides and included angle:

\text{Area} = \frac{1}{2}ab\sin C

2. Basic Trigonometric Ratios in Right Triangles

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

3. Inradius (r) and Circumradius (R)

For all triangles:

\text{Inradius} = \frac{A}{s}

For right triangle:

\text{Inradius} = \frac{a + b - c}{2}

Circumradius for right triangle:

R = \frac{c}{2} \quad (\text{hypotenuse})

Visual Explanation

Right Triangle with Height

       C
      /|
     / |
  b /  | a
   /   |
  /____|
A       B
     c

Right angle at C
AB = hypotenuse = c
Area = $\frac{1}{2}ab$

Conceptual Tips and Common Mistakes

Mistake	Tip
Using wrong base-height pair	Base and height must be perpendicular
Misapplying Pythagoras to non-right triangles	Use only in right-angled triangles
Ignoring the triangle inequality while constructing triangle	Always check side conditions
Assuming all triangles have inradius/circumradius formulas same as equilateral	Each triangle type has specific formulas

Shortcuts and Tricks

For an isosceles triangle, draw height from apex to split into two right triangles.
For quick MCQs, use the 180° angle sum and exterior angle properties.
Max area triangle for fixed perimeter is equilateral.

Examples

Example 1:

Triangle has sides 5, 6, 7. Find area.

Solution:
Use Heron’s Formula:

s = \frac{5+6+7}{2} = 9 \\ \text{Area} = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \cdot 4 \cdot 3 \cdot 2} = \sqrt{216} \approx 14.7

Example 2:

Find third angle of triangle with angles 65°, 45°.

\angle C = 180^\circ - (65^\circ + 45^\circ) = 70^\circ

Example 3:

A triangle has sides 13, 12, and 5. Is it right-angled?

Check:

5^2 + 12^2 = 25 + 144 = 169 = 13^2 \Rightarrow \text{Yes}