Let H be the orthocenter of an acute - angled triangle ABC and O be itscircumcenter. Then `vec(HA)+vec(HB)+vec(HC)`

Question

Let H be the orthocenter of an acute - angled triangle ABC and O be itscircumcenter. Then `vec(HA)+vec(HB)+vec(HC)`

Let H be the orthocenter of an acute - angled triangle ABC and O be itscircumcenter. Then `vec(HA)+vec(HB)+vec(HC)`
A. is equal to `vec(HO)`
B. is equal to `vec(3HO)`
C. is equal to `vec(2HO)`
D. is not a scalar multiple of `vec(HO)` in general

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Salim Ahmed Kawsar

Correct Answer - C
G ic centroid
`G=(A+B+C)/(3)`
`G=(20+H)/(3)`
`2O+H=3G`
`vec(HA)+vec(HB)+vec(HC)=vec(A)-vec(H)-vec(B)+vec(C)-vec(H)`
`=vec(A)+vec(B)+vec(C)-3vec(H)`
`=3vec(G)-3vec(H)`
`=2vec(O)+vecH-3vec(H)`
`=2vec(O)-2vec(H)`
`=2vec(HO)`

4 views Answered 2023-02-05 15:29