A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #( pi)/4 #, and the triangle's area is #18 #. What is the area of the triangle's incircle?

Answer 1

#~~6.97#

The third angle of the triangle is #/_C = pi -pi/4-pi/8={5 pi}/8#

Sine law gives us

#a/{sin (pi/8)} = b/{sin(pi/4)}=c/{sin({5 pi}/8)} = k ( "say")#

The area is given by

#A = 1/2 b c sin/_A = 1/2 k^2 sin/_A sin/_B sin/_C = 1/2 k^2 sin(pi/8) sin(pi/4)sin({5pi}/8) = 1/2 k^2 sin(pi/8) sin(pi/4) cos(pi/4) = 1/4 k^2 sin^2(pi/4) = 1/8 k^2#
and thus #k^2 = 8 xx 18 = 144# or #k=12#.

Now, the radius of the incircle can be easily found from

#A = 1/2 (a+b+c)r#

which one can derive simply by joining each of the vertices with the incenter. So, we have

#1/2 k(sin(pi/8) +sin(pi/4)+sin({5pi}/8))r = 18#

and so

#r = 6/(sin(pi/8) +sin(pi/4)+sin({5pi}/8)) ~~ 1.49 #

Thus the area of the incircle is

#pi r^2 ~~ 6.97#
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Answer 2

To find the area of the triangle's incircle, we can use the formula:

Area of incircle = Semiperimeter of triangle * Inradius

First, we need to find the semiperimeter of the triangle, which is the half of the perimeter:

Semiperimeter = (AB + BC + AC) / 2

Next, we can use Heron's formula to find the area of the triangle:

Area of triangle = √(s(s - AB)(s - BC)(s - AC))

where s is the semiperimeter, AB, BC, and AC are the lengths of the sides of the triangle.

Once we have the area of the triangle, we can find the inradius using the formula:

Inradius = Area of triangle / Semiperimeter

Then, we can use the inradius to find the area of the incircle using the formula:

Area of incircle = π * (Inradius)^2

By substituting the given values and calculating accordingly, we can find the area of the triangle's incircle.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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