Math, asked by rahulbuchadi6, 2 months ago

LMN is an isosceles triangle in which LM = LN and (ZL) = 2(2M). LM = 4 cm. what is the ratio of inradius to the circumradius?​

Answers

Answered by amitnrw
0

Given : LMN is an isosceles triangle in which LM = LN and (∠L) = 2(∠M). LM = 4 cm.

To Find : ratio of inradius to the circumradius

Solution:

LM = LN

=> ∠M = ∠N

∠L = 2∠M

=> ∠L =∠M  +  ∠N

∠L  + ∠M + ∠N = 180°   ( sum of angles of triangle)

=> 2 ∠L = 180°  

=>  ∠L = 90°  

This is isosceles right angle triangle

LM =LN = 4 cm

=> MN = 4√2    (∵ MN² =  LM² + LN²)

Circumradius of right angle triangle = (1/2) hypotenuse  =

= (1/2)  4√2  

= 2√2   cm

Perimeter of triangle = 4 + 4 + 4√2   = 4 (2 +√2)cm

Area of triangle = (1/2) * 4 * 4  = 8  cm²

Area of triangle = (1/2)(perimeter ) * in radius

=> (1/2)(4 (2 +√2) * in radius  = 8

in radius =  4/ (2 +√2)

ratio of inradius to the circumradius  =   4/ (2 +√2)2√2

= 4/(4√2 + 4)

= 1/(√2 + 1)  

= √2 -  1

ratio of inradius to the circumradius  =   1 : (√2 + 1)

or (√2 -  1)  : 1

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