AP EAMCET · Maths · Properties of Triangles

In $∆ A B C$ , suppose the radius of the circle opposite to angle $A, B$ and $C$ is denoted by $r_{1}$ , $r_{2}$ and $r_{3}$ . If $r_{1} = 2, r_{2} = 3, r_{3} = 6$ and $R$ is the radius of the circumcircle, then the value of $r_{1} + r_{2} + r_{3} - r =$

A $4 R$
B $3 R$
C $2 R$
D $R$

Verified Solution

Answer & Solution

Correct Answer

(A) $4 R$

Step-by-step Solution

Detailed explanation

For △ABC, exradii r1=△s-a, r2=△s-b, r1=△s-c and inradius r=△s, where s is the semi-perimeter of the triangle and r is the radius of in-circle. And, △=s(s-a)(s-b)(s-c)=abc4R Now, r1+r2+r3-r =△s-a+△s-b+△s-c-△s…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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In ∆ABC, suppose the radius of the circle opposite to angle A, B and C is denoted by r1, r2 and r3. If r1=2, r2=3, r3=6 and R is the radius of the circumcircle, then the value of r1+r2+r3-r=

Step-by-step Solution

In $∆ A B C$ , suppose the radius of the circle opposite to angle $A, B$ and $C$ is denoted by $r_{1}$ , $r_{2}$ and $r_{3}$ . If $r_{1} = 2, r_{2} = 3, r_{3} = 6$ and $R$ is the radius of the circumcircle, then the value of $r_{1} + r_{2} + r_{3} - r =$