Triangle Geometry Theorems and Formulas: Centres, Similar Triangles and Right Triangle Rules for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
In this note
PDF Notes: Triangle Geometry Theorems and Formulas: Centres, Similar Triangles and Right Triangle Rules for SSC
Premium PDF · one-time payment of ₹5 · instant access
Geometry questions in SSC CGL are mostly on triangles and circles, and the triangle questions use about ten theorems. This triangle geometry theorems and formulas guide for SSC collects the angle properties, the four centres, the right-triangle formulas, similar triangles and the basic proportionality theorem, with solved examples.
Angle Properties
- The sum of the angles of a triangle is 180°.
- The exterior angle is equal to the sum of the two interior opposite angles.
- In a triangle, the larger side is opposite to the larger angle.
- The sum of any two sides is more than the third side.
Four Centres of a Triangle
| Centre | Made by | Property |
|---|---|---|
| Centroid (G) | Meeting point of the medians | Divides each median in the ratio 2 : 1 |
| Incentre (I) | Meeting point of the angle bisectors | Equidistant from the three sides; the centre of the incircle |
| Circumcentre (O) | Meeting point of the perpendicular bisectors of the sides | Equidistant from the three vertices; the centre of the circumcircle |
| Orthocentre (H) | Meeting point of the altitudes | In a right triangle it is at the right-angle vertex |
Important Formulas
| Item | Formula |
|---|---|
| Area (Heron) | √[s(s − a)(s − b)(s − c)] |
| Inradius r | Area / s (s = semi-perimeter) |
| Circumradius R | abc / (4 × Area) |
| Right triangle: circumradius | Half of the hypotenuse |
| Right triangle: inradius | (a + b − c) / 2, where c is the hypotenuse |
| Equilateral triangle (side a): inradius | a / (2√3) |
| Equilateral triangle (side a): circumradius | a / √3 |
| Pythagoras | Hypotenuse² = base² + height² |
Similar Triangles and BPT
- Similar triangles have equal angles and sides in the same ratio.
- The ratio of the areas of similar triangles is the square of the ratio of the sides.
- Basic proportionality theorem (BPT): if DE is parallel to BC in triangle ABC, then AD/DB = AE/EC.
- Midpoint theorem: the line joining the midpoints of two sides is parallel to the third side and is half of it.
Solved Examples
Example 1. Two interior opposite angles are 50° and 60°. Find the exterior angle.
50 + 60 = 110°.
Example 2. The sides of a right triangle are 3, 4 and 5. Find the inradius.
(3 + 4 − 5)/2 = 1.
Example 3. The hypotenuse of a right triangle is 10. Find the circumradius.
Half of 10 = 5.
Example 4. The sides of two similar triangles are in the ratio 2 : 3. Find the ratio of their areas.
2² : 3² = 4 : 9.
Example 5 (BPT). In triangle ABC, DE is parallel to BC. AD = 4, DB = 6 and AE = 2. Find EC.
4/6 = 2/EC, so EC = 3.
Example 6. The side of an equilateral triangle is 6. Find the circumradius.
6/√3 = 2√3.
Quick Tips
- Learn the four centres and their properties.
- The right triangle has the circumcentre at the midpoint of the hypotenuse.
- Use the square of the ratio for the areas of similar triangles.
- Write the BPT ratio with the same order of letters.
Quick Revision Points
- The angle sum of a triangle is 180°.
- The centroid divides the median in 2 : 1.
- The circumradius of a right triangle is half of the hypotenuse.
- The area ratio of similar triangles is the square of the side ratio.
- BPT: AD/DB = AE/EC.
Practice Questions With Answers
- What is the sum of the angles of a triangle?
Answer: 180° - Two interior opposite angles are 50° and 60°. Find the exterior angle.
Answer: 110° - Find the hypotenuse of a right triangle with sides 5 and 12.
Answer: 13 - The hypotenuse of a right triangle is 10. Find the circumradius.
Answer: 5 - Find the inradius of a right triangle with sides 3, 4 and 5.
Answer: 1 - In what ratio does the centroid divide a median?
Answer: 2 : 1 - The sides of two similar triangles are in the ratio 2 : 3. Find the ratio of their areas.
Answer: 4 : 9 - DE is parallel to BC, AD = 4, DB = 6, AE = 2. Find EC.
Answer: 3 - Find the circumradius of an equilateral triangle of side 6.
Answer: 2√3 - Where is the orthocentre of a right triangle?
Answer: At the vertex of the right angle
How to Practise This Chapter
Write each formula on a card, with a solved example on the back, and revise the cards every day. Then test yourself with our practice papers and the previous year papers page. See all guides on the Formula Guide page, and also read the Mathematics notes and Quantitative Aptitude notes. The previous guide is Height and Distance and the next guide is Circle Geometry Theorems.
Frequently Asked Questions
What is the centroid of a triangle?
It is the meeting point of the three medians, and it divides each median in the ratio 2 : 1.
What is the incentre?
It is the meeting point of the angle bisectors, and it is the centre of the incircle.
What is the circumcentre?
It is the meeting point of the perpendicular bisectors of the sides, and the centre of the circumcircle.
What is the orthocentre?
It is the meeting point of the altitudes.
What is the exterior angle theorem?
The exterior angle equals the sum of the two interior opposite angles.
What is the formula of the area ratio of similar triangles?
It is the square of the ratio of the corresponding sides.
What is BPT?
If a line is parallel to one side of a triangle, it divides the other two sides in the same ratio.
What is the circumradius of a right triangle?
Half of the hypotenuse.
Is geometry asked in SSC exams?
Yes. Geometry is asked in SSC CGL maths, with 5 to 7 questions.
How much time should I take?
About 40 to 60 seconds.
What are common mistakes?
Mixing up the centres and forgetting to square the ratio for areas.
How can I learn the theorems?
Draw a figure for each theorem and write the result next to it.
Previous chapter
Height and Distance Formula Guide: Angle of Elevation, Depression and Two-Angle Questions for SSC
Next chapter
Circle Geometry Theorems and Formulas: Chords, Tangents, Cyclic Quadrilateral and Common Tangents for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.