The Triangle and Its Properties
"The triangle is the STRONGEST geometric shape — you will find it in bridges, roofs, and even the pyramids of Egypt!"
1. What Is a Triangle?
A triangle is a polygon with THREE sides, THREE vertices, and THREE angles. Notation: ΔABC — vertices A, B, C; sides AB, BC, CA; angles ∠A, ∠B, ∠C.
Classification by Sides
| Type | Sides | Example | Properties |
|---|---|---|---|
| Scalene | All sides DIFFERENT | 3 cm, 4 cm, 5 cm | All angles different |
| Isosceles | TWO sides EQUAL | 5 cm, 5 cm, 3 cm | Base angles EQUAL |
| Equilateral | ALL sides EQUAL | 4 cm, 4 cm, 4 cm | Each angle = 60° |
Classification by Angles
| Type | Angle | Example | Property |
|---|---|---|---|
| Acute-angled | ALL angles < 90° | 60°, 50°, 70° | All three acute |
| Right-angled | ONE angle = 90° | 90°, 45°, 45° | Pythagoras theorem applies |
| Obtuse-angled | ONE angle > 90° | 110°, 35°, 35° | Only one obtuse possible |
2. Medians of a Triangle
Median: A line segment from a VERTEX to the MIDPOINT of the OPPOSITE side. Every triangle has THREE medians.
'Think of the median as the "middle line" — it connects the vertex to the exact centre of the opposite side.'
Properties:
- A median divides the triangle into TWO triangles of EQUAL area.
- All THREE medians intersect at a point called the CENTROID (also called the centre of gravity).
- The centroid divides each median in the ratio 2:1 (vertex to centroid : centroid to midpoint).
3. Altitudes of a Triangle
Altitude: A perpendicular line segment from a VERTEX to the OPPOSITE side (or its extension). Also called the height. Every triangle has THREE altitudes.
| Triangle Type | Altitude Location |
|---|---|
| Acute-angled | ALL altitudes lie INSIDE the triangle |
| Right-angled | TWO altitudes are the legs; the third is inside |
| Obtuse-angled | ONLY one altitude lies INSIDE; two lie OUTSIDE |
Orthocentre: The point where all three altitudes intersect.
'Do NOT confuse median with altitude. Median goes to the MIDPOINT. Altitude is PERPENDICULAR to the opposite side — they are the same ONLY in an equilateral or isosceles (when drawn to the base) triangle.'
4. Exterior Angle Property
Exterior Angle: Formed when a side of the triangle is EXTENDED. It forms a LINEAR PAIR with the adjacent interior angle.
Exterior Angle Theorem
'An EXTERIOR angle of a triangle is EQUAL to the SUM of the TWO INTERIOR OPPOSITE angles.'
Mathematically: ∠ACD (exterior) = ∠A + ∠B, where A and B are the two interior opposite angles.
Worked Example: In ΔABC, ∠A = 50°, ∠B = 65°. Find the exterior angle at C. Exterior angle = ∠A + ∠B = 50° + 65° = 115°. Check: Interior ∠C = 180° − (50° + 65°) = 65°. Exterior + Interior = 115° + 65° = 180° ✓.
Why This Matters: 'The exterior angle theorem gives a QUICK way to find an unknown angle without computing all interior angles first.'
5. Angle Sum Property
Angle Sum of a Triangle: The sum of the THREE interior angles is ALWAYS 180°.
Proof by paper folding: 'Draw a triangle, cut it out, tear off the three corners, and place them together. They form a STRAIGHT LINE — 180°.'
Worked Example: Two angles of a triangle are 45° and 75°. Find the third. Third angle = 180° − (45° + 75°) = 180° − 120° = 60°.
Application: 'In a right triangle, the two acute angles are COMPLEMENTARY (sum = 90°).'
6. Triangle Inequality
Theorem: The SUM of any TWO sides of a triangle is GREATER than the THIRD side.
| Side Triplet | Check | Valid Triangle? |
|---|---|---|
| 3, 4, 5 | 3+4 > 5 ✓, 4+5 > 3 ✓, 3+5 > 4 ✓ | YES |
| 2, 3, 6 | 2+3 = 5 < 6 | NO |
| 5, 5, 9 | 5+5 = 10 > 9 ✓, others ✓ | YES |
| 1, 2, 3 | 1+2 = 3, NOT greater | NO |
'This property tells us if three given lengths can form a triangle. It is used in construction — engineers always check this before building triangular structures.'
7. Equilateral and Isosceles Triangles — Special Properties
Equilateral Triangle
- ALL sides equal. ALL angles = 60°.
- Each median = each altitude.
- The centroid, orthocentre, incentre, and circumcentre ALL coincide.
Isosceles Triangle
- TWO sides equal (called the LEGS). Third side = BASE.
- Angles opposite equal sides are EQUAL (base angles).
- 'The altitude from the apex to the base also acts as the MEDIAN and ANGLE BISECTOR.'
8. Right Triangles and Pythagoras Theorem
'For a RIGHT-ANGLED triangle, Pythagoras discovered a beautiful relationship between the three sides.'
Pythagoras Theorem: In a right triangle, the SQUARE of the HYPOTENUSE is EQUAL to the SUM OF SQUARES of the other TWO sides.
Formula: (Hypotenuse)² = (Base)² + (Height)² or c² = a² + b², where c is the hypotenuse.
Hypotenuse: The side OPPOSITE the right angle — ALWAYS the LONGEST side.
Worked Examples
Example 1: Find the hypotenuse if base = 3 cm, height = 4 cm. c² = 3² + 4² = 9 + 16 = 25. c = √25 = 5 cm.
Example 2: A ladder of length 13 m reaches a window 12 m above ground. How far is the foot of the ladder from the wall? Let distance = x. 13² = 12² + x² → 169 = 144 + x² → x² = 25 → x = 5 m.
Example 3: Check if a triangle with sides 7, 24, 25 is right-angled. Longest side = 25. Check: 7² + 24² = 49 + 576 = 625 = 25². YES, it is a right triangle.
Pythagorean Triplets
Sets of three positive integers (a, b, c) that satisfy a² + b² = c². Common triplets: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41). 'If you multiply a Pythagorean triplet by the SAME number, you get ANOTHER triplet. Example: (3, 4, 5) × 2 = (6, 8, 10) — also works!'
9. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Any three numbers can be sides of a triangle | Violates triangle inequality | Sum of ANY two must be > third |
| Exterior angle = sum of ALL interior opposite angles | Not ALL — only the two NON-adjacent interior angles | Only the two opposite interior angles |
| Hypotenuse is always side 'c' | It is the side OPPOSITE the right angle, not the third alphabetically | Identify the right angle first |
| 2, 3, 4 cannot form a triangle | It CAN — 2+3 > 4, 3+4 > 2, 2+4 > 3 | Actually valid! |
10. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Angle sum property | 2-3 | Find unknown angle |
| Exterior angle theorem | 2-3 | Word problems |
| Triangle inequality | 2-3 | Valid triangle check |
| Pythagoras theorem | 3-4 | Computation and word problems |
| Median and altitude | 1-2 | Definitions |
Quick Self-Test
Q1. Find the third angle of a triangle with angles 35° and 85°. A1. 180° − (35° + 85°) = 180° − 120° = 60°.
Q2. Is a triangle with sides 4, 7, 12 possible? A2. 4 + 7 = 11 < 12. NO, triangle inequality is violated.
Q3. Find the hypotenuse of a right triangle with legs 6 cm and 8 cm. A3. h² = 6² + 8² = 36 + 64 = 100. h = 10 cm.
Q4. An exterior angle of a triangle is 110° and one interior opposite angle is 45°. Find the other. A4. Exterior = sum of two opposite. 110° = 45° + x → x = 65°.
Q5. A ladder 10 m long reaches a window 8 m high. Find the distance of the foot from the wall. A5. 10² = 8² + x² → 100 = 64 + x² → x² = 36 → x = 6 m.
Q6. In an isosceles triangle, base angles are 50° each. Find the vertex angle. A6. Vertex angle = 180° − (50° + 50°) = 80°.
Q7. Check if (12, 35, 37) is a Pythagorean triplet. A7. 12² + 35² = 144 + 1225 = 1369 = 37². YES, it is a triplet.
