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    inscribed circle

    the sides of a triangle are 80 cm, 100cm, and 140 cm. determine the radius of the inscribed circle
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    Quote Originally Posted by aeroflix View Post
    the sides of a triangle are a = 80 cm, b = 100cm, and c = 140 cm. determine the radius of the inscribed circle
    1. Draw a sketch.

    2. Calculate the area of the triangle: Use Cosine rule to determine one interior angle (i.e. \alpha) of the triangle, for instance the angle opposite the side a. Then the area is:

    A = \frac12 \cdot c \cdot b \cdot \sin(\alpha)

    3. The center of the inscribed circle is the common vertex of three interior triangles whose area is:

    \frac12 \cdot a \cdot r + \frac12 \cdot b \cdot r + \frac12 \cdot c \cdot r = A

    \frac12 \cdot (a+b+c) \cdot r = A~\implies~\boxed{r = \dfrac{2A}{a+b+c}}
    Attached Thumbnails Attached Thumbnails inscribed circle-rhoaus3eckfl.png  
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  3. #3
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    Quote Originally Posted by aeroflix View Post
    the sides of a triangle are 80 cm, 100cm, and 140 cm. determine the radius of the inscribed circle
    Here comes a different approach to solve this question:

    1. Use Cosine rule to determine the interior angles of the triangle.

    2. Let denote \alpha the angle at A and \beta the angle at B. Then you know:

    \underbrace{\dfrac r{\tan\left(\frac{\alpha}{2} \right)}}_{x} + \underbrace{\dfrac r{\tan\left(\frac{\beta}{2} \right)}}_{y} = c

    3. Solve for r:

    r = c \cdot \dfrac{\tan\left(\frac{\alpha}{2} \right) \cdot \tan\left(\frac{\beta}{2} \right)}{\tan\left(\frac{\alpha}{2} \right) + \tan\left(\frac{\beta}{2} \right)}
    Attached Thumbnails Attached Thumbnails inscribed circle-rhoaushalbwinkl.png  
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