Question

in ∆ABC, a=3,b=4,c=5 , then find distance between incentre and circumcentre​

Answer 1

Question :- in ∆ABC, a = 3,b = 4,c = 5 , then find distance between incentre and circumcentre ?

Solution :-

Euler’s Theorem in geometry, states that the distance between the incentre and circumcentre of a triangle is :-

• √(R² - 2Rr) where,
• R = circumradius of the ∆ .
• r = inradius of the ∆ .

sides of given ∆'s are 3 , 4 and 5 unit.

As we know,

→ 3² + 4² = 5²

→ 9 + 16 = 25

→ 25 = 25 .

Therefore, we can conclude that, given ∆ is a right angle ∆.

Now, in Right angle ∆ :-

• inradius = (Perpendicular + Base - Hypotenuse) / 2 .
• circumradius = (Hypotenuse / 2) .

So,

Putting values we get :-

→ inradius of given right ∆ = (3 + 4 - 5)/2 = (7 - 5)/2 = 2/2 = 1

→ circumradius = 5/2 = 2.5

Putting both values in Euler’s Theorem , we get,

→ Distance = √(2.5² - 2*2.5*1)

→ Distance = √(6.25 - 5)

→ Distance = √(1.25)

→ Distance ≈ 1.1 unit. (Ans.)

Hence, the distance between incentre and circumcentre is 1.1 unit.

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किसी त्रिभुज का आधार एवं संगत शिर्शलंब क्रमश (x+y)^2 एवं (x-y)^2 है तो उसका च छेत्रफल क्या होगी ।

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