O is any point in the interior of - ABC- Prove that i-AB-AC-OB-OC ii-AB-BC-CA-OA-OB-OC iii-OA-OB-OC-1-2AB-BC-CA

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Triangle Inequality

O is any poin...

Question

O is any point in the interior of $Δ A B C .$

Prove that

$(i) A B + A C > O B + O C$

$(i i) A B + B C + C A > O A + O B + O C$

$(i i i) O A + O B + O C > \frac{1}{2} (A B + B C + C A)$

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Solution

Given : In $Δ A B C, O is any point in the interior of the Δ A B C, OA, OB and OC are joined$

To Prove :

$(i) A B + A C > O B + O C$

$(i i) A B + B C + C A > O A + O B + O C$

$(i i i) O A + O B + O C > \frac{1}{2} (A B + B C + C A)$

Construction : Produce BO to meet AC in D.

Proof : In $Δ A B D,$

(i) $A B + A D > B D$

(Sum of any two sides of a triangle is greater than thrid)

$\Rightarrow A B + A D > B O + O D . . . (i) S i m i l a r l y, i n Δ O D C, O D + D C > O C . . . (i i) Adding (i) and (ii) A B + A D + O D + D C > O B + O D + O C \Rightarrow A B + A D + D C > O B + O C \Rightarrow A B + A C > O B + O C (ii)~Similarly, we can prove that B C + A B + O A + O C a n d C A + B C > O A + O B (ii) I n Δ O A B, Δ O B C a n d Δ O C A, O A + O B > A B O B + O C > B C a n d O C + O A > C A Adding, we get 2 (O A + O B + O C) > A B + B C + C A ∴ O A + O B + O C > \frac{1}{2} (A B + B C + C A)$

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Similar questions

$If O is a point within Δ A B C, show that (i) AB + AC > O B + O C (ii) AB + BC + CA > O A + O B + O C (iii) OA + OB + OC > \frac{1}{2} (A B + B C + C A .)$

. O is any point in the interior of a triangle ABC. Prove that

1. AB + AC > OB + OC

2.AB + BC + CA > OA + OB + OC

3.OA + OB + OC > 1/2(AB + AC + BC)

Q. If

O

is a point within

Δ

ABC then show that :
1)

A B + A C = O B + O C

A B + B C + C A > O A + O B + O C

O A + O B + O C > \frac{1}{2} (A B + B C + C A)

If $O$ is a point within triangle $A B C$ , show that,

Q. O is any point in the interior of ∆ABC. Then, which of the following is true?
(a)

(O A + O B + O C) > (A B + B C + C A)

(b)

(O A + O B + O C) > \frac{1}{2} (A B + B C + C A)

(c)

(O A + O B + O C) < \frac{1}{2} (A B + B C + C A)

(d) None of these

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O is any point in the interior of ΔABC. Prove that (i) AB+AC>OB+OC (ii) AB+BC+CA>OA+OB+OC (iii) OA+OB+OC>12(AB+BC+CA)

O is any point in the interior of $Δ A B C .$

Prove that

$(i) A B + A C > O B + O C$

$(i i) A B + B C + C A > O A + O B + O C$

$(i i i) O A + O B + O C > \frac{1}{2} (A B + B C + C A)$