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Find the inradius of trianlge ABC.

 

Find the circumradius of triangle ABC.

 

 Dec 18, 2023

Best Answer 

 #1
avatar+259 
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Sure, I can help you find the inradius and circumradius of triangle ABC with sides 1, sqrt(2), and sqrt(2).

 

Inradius:

 

Calculate the semiperimeter (s) of the triangle:

 

s = (a + b + c) / 2 = (1 + sqrt(2) + sqrt(2)) / 2 = 1 + sqrt(2)

 

Use Heron's formula to find the area (K) of the triangle:

 

K = sqrt(s * (s - a) * (s - b) * (s - c))\

 

K = sqrt((1 + sqrt(2)) * (0 + sqrt(2)) * (1 + sqrt(2)) * (0 + sqrt(2)))

 

K= sqrt(2^(3 + 1/2)) = 2 * sqrt(2^(1/2)) = 2

 

Finally, calculate the inradius (r) using the formula:

 

r = K / s r = 2 / (1 + sqrt(2))

 

Circumradius:

 

Use the Law of Cosines to find the angle A of the triangle:

 

cos(A) = (b^2 + c^2 - a^2) / (2bc

 

cos(A) = ((sqrt(2))^2 + (sqrt(2))^2 - 1^2) / (2 * sqrt(2) * sqrt(2))

 

cos(A) = 0 A = pi/2

 

Since angle A is a right angle, the circumradius (R) is simply half the hypotenuse:

R = a / 2 R = 1 / 2

 

Therefore, the inradius of triangle ABC is 2 / (1 + sqrt(2)) and the circumradius is 1 / 2.

 Dec 18, 2023
 #1
avatar+259 
+1
Best Answer

Sure, I can help you find the inradius and circumradius of triangle ABC with sides 1, sqrt(2), and sqrt(2).

 

Inradius:

 

Calculate the semiperimeter (s) of the triangle:

 

s = (a + b + c) / 2 = (1 + sqrt(2) + sqrt(2)) / 2 = 1 + sqrt(2)

 

Use Heron's formula to find the area (K) of the triangle:

 

K = sqrt(s * (s - a) * (s - b) * (s - c))\

 

K = sqrt((1 + sqrt(2)) * (0 + sqrt(2)) * (1 + sqrt(2)) * (0 + sqrt(2)))

 

K= sqrt(2^(3 + 1/2)) = 2 * sqrt(2^(1/2)) = 2

 

Finally, calculate the inradius (r) using the formula:

 

r = K / s r = 2 / (1 + sqrt(2))

 

Circumradius:

 

Use the Law of Cosines to find the angle A of the triangle:

 

cos(A) = (b^2 + c^2 - a^2) / (2bc

 

cos(A) = ((sqrt(2))^2 + (sqrt(2))^2 - 1^2) / (2 * sqrt(2) * sqrt(2))

 

cos(A) = 0 A = pi/2

 

Since angle A is a right angle, the circumradius (R) is simply half the hypotenuse:

R = a / 2 R = 1 / 2

 

Therefore, the inradius of triangle ABC is 2 / (1 + sqrt(2)) and the circumradius is 1 / 2.

BuiIderBoi Dec 18, 2023

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