Take any point o in the interior of
triangle ABC as shown.
State which of the following is true?
(a) AO + BO > AB
(b) BO + CO > BC
(c) CO+AO CA
(d) AB + ACBC
(e) AB + BC + CA 2(AO + BO + CO)
Answers
Answer:The sum of any two sides of triangle is always greater than third
[A] In ∆ ABF
AB + AF > BF
AB + AF > BO + OF .....(a)
In ∆ COF
CF + FO > OC .....(b)
add both equation a & b
AB + AF + CF + FO > BO + OF + OC
Substitute "OF"
AB + AF +FC > BO + OC
AB + AC > BO + OC ......(1)
[B] In ∆ BFC
FC + BC > BF
FC + BC > BO + OF ......(c)
In ∆ AFO
OF + AF >AO .....(d)
Add equation c & d
FC + BC + OF + AF > BO + OF + AO
Substitute " OF "
FC + AF + BC > BO + AO
AC + BC > BO + AO ......(2)
[C] In ∆ EBC
EB + BC > EC
EB + BC > EO + OC....(e)
In ∆ AEO
EA + EO > AO....(f)
add both equation e & f
EB + BC + EA + EO > EO + OC + AO
Substitute " EO "
(EB + EA) + BC > AO +OC
AB + BC > OA + OC ......(3)
add equation 1 , 2 & 3
2AB + 2AC + 2BC > 2BO + 2OC +2OA
2(AB + AC + BC) > 2(BO + OC + OA)
FOLLOW ME
Answer:
(a) True
(b) True
(c) True
(d) False
(e) True