Geometry · Topic 3 of 7

🎯 Triangle Centres

3 exam question types, fully solved

Every triangle hides three famous “centres”, and SSC loves them because each one comes with a single memorised rule that turns a scary-looking figure into a one-line answer. The trick is never to confuse them — the centroid, the incentre and the circumcentre are built from different lines and obey different formulas.

The three centres and their one-line rules

• Centroid (G): where the three medians meet. It divides every median in the ratio from the vertex, so the vertex piece is of the median and the midpoint piece is .

• Incentre (I): where the three internal angle bisectors meet. It is equidistant from all three sides — that distance is the inradius , with Area . Its key angle: .

• Circumcentre (O): where the three perpendicular bisectors of the sides meet. It is equidistant from all three vertices — that distance is the circumradius (). Its key angle: .

Two more memory hooks: for a right triangle the circumcentre is the midpoint of the hypotenuse, so ; and the incentre angle is always obtuse while the circumcentre central angle is double the vertex angle. The fatal exam mistake is swapping (incentre) with (circumcentre).

🎯Triangle Centres LabPick a triangle shape, then a centre — real construction lines, live angle check

∠A = 65°∠B = 55°

∠C = 180° − 65° − 55° = 60° · acute triangle

Centroid — medians meet, 2 : 1 split

A median joins a vertex to the midpoint of the opposite side.

Measured on median B: vertex→G = 2 × G→midpoint

AG : GM = 2 : 1 (vertex side is always double) ✓

Using the incentre rule : if, then .

Keep the two angle laws side by side: the incentre angle grows from a base of , while the circumcentre angle simply doubles the vertex angle. Decide which centre the question names before you pick a formula.

Q1In △ABC, G is the centroid and median AD = 24 cm. Find GD.

Q2I is the incentre of △ABC with ∠A = 70°. Find ∠BIC.

Q3O is the circumcentre of △ABC and ∠A = 50°. Find ∠BOC.

📚

Real exam questions — Triangle Centres

3 question types · 15 solved examples from real SSC papers

Each centre comes from one construction and one fact: the centroid splits medians 2:1, the incentre gives the 90 + half-A angle, and the circumcentre doubles the vertex angle.

How to solve this type

A median joins a vertex to the midpoint of the opposite side; all three medians cross at one point, the centroid

G

. The single fact that cracks most questions:

G

divides each median in the ratio

2:1

measured from the vertex, so the vertex piece is the longer

\dfrac{2}{3}

and the midpoint piece is the shorter

\dfrac{1}{3}

. Two more handy results: when two medians are perpendicular, the side between their vertices is

\dfrac{2}{3}\sqrt{m_1^2+m_2^2}

, and the three medians slice the triangle into 6 small triangles of equal area. The #1 trap is reversing the ratio — always give the bigger share to the vertex.

G is the centroid of $\triangle ABC$ and AD is a median with $AD = 18$ cm. Find AG.

6

cmB

15

cmC

12

cmD

9

Two medians of a triangle, of lengths 12 cm and 9 cm, are perpendicular to each other. The side joining their two vertices is:

2\sqrt{16+36}

cmB

10

cmC

2\sqrt{64+36}

cmD

10\sqrt{2}

In $\triangle ABC$ , G is the centroid. If the area of $\triangle ABG = 12$ cm $^2$ , find the area of $\triangle ABC$ .

36

24

48

30

The three medians of a triangle are 9 cm, 12 cm and 15 cm. The sum of the distances from the centroid to the three vertices is:

24

cmB

30

cmC

12

cmD

18

The centroid G divides median AM ( $AM = 21$ cm) in the ratio $2:1$ . Find GM, the distance from G to the midpoint M.

10.5

cmB

7

cmC

14

cmD

5