FORMULA CHAPTER
Perpendicular bisectors of AB, BC and CA meet at O
OA = OB = OC
AD ⟂ BC, BE ⟂ CA and CF ⟂ AB meet at H
Internal bisectors of ∠A, ∠B and ∠C meet at I
Medians AD, BE and CF meet at G
AG : GD = BG : GE = CG : GF = 2 : 1
Perpendicular bisectors of AB, BC and CA meet at O
OA = OB = OC
Question: The perpendicular bisectors of AB and BC of triangle ABC meet at O. If OA = 6 cm, find OB and OC.
Calculation:
Step 1: The perpendicular bisectors are concurrent at circumcentre O.
Step 2: A circumcentre is equidistant from all three vertices.
Step 3: OA = OB = OC = 6 cm.
Answer: OB = 6 cm and OC = 6 cm.
AD ⟂ BC, BE ⟂ CA and CF ⟂ AB meet at H
Question: In triangle ABC, AD ⟂ BC and BE ⟂ CA meet at H. Through which point does the third altitude from C pass?
Calculation:
Step 1: AD and BE are two altitudes.
Step 2: By the altitude concurrency theorem, all three altitudes meet at one point.
Step 3: Therefore the altitude from C also passes through H.
Answer: The third altitude passes through H, the orthocentre.
Internal bisectors of ∠A, ∠B and ∠C meet at I
Question: The internal bisectors of ∠A and ∠B of triangle ABC meet at I. Where does the internal bisector of ∠C pass?
Calculation:
Step 1: I is the intersection of two internal angle bisectors.
Step 2: All three internal angle bisectors of a triangle are concurrent.
Step 3: Therefore the bisector of ∠C also passes through I.
Answer: It passes through I, the incentre.
Medians AD, BE and CF meet at G
AG : GD = BG : GE = CG : GF = 2 : 1
Question: In triangle ABC, median AD passes through centroid G. If GD = 4 cm, find AG and AD.
Calculation:
Step 1: A centroid divides a median in the ratio AG : GD = 2 : 1.
Step 2: AG = 2 × GD = 2 × 4 = 8 cm.
Step 3: AD = AG + GD = 8 + 4 = 12 cm.
Answer: AG = 8 cm and AD = 12 cm.
