ln a △ A B C , A = 30 ° + C and R = ( 3 + 1 ) r , where r is the inradius and R is the…

ln a △ABC, A=30°+C and R=(3+1)r, where r is the inradius and R is the circumradius, then
  1. ABC is a right-angled triangle
  2. ABC is an equilateral triangle
  3. ABC is acute angled
  4. ∠A=75°, ∠B=60°, ∠C=45°

Solution

Using

r=4RsinA2·sinB2·sinC2

⇒R3+1=4RsinA2·sinB2·sinC2 ∵R=3+1r

⇒3-18=sinA2·sinB2·sinC2

⇒sinA2·sinB2·sinC2=3-122·12·12

Now, A2, B2, C2 are 12, 12, 3-122 and it is given that A=30°+C 

So, we can say that A, B & C are 60°, 90° & 30° respectively

Asked in: AP EAMCET 2020 (23 Sep Shift 1)

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