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