# A triangle has sides with lengths of 8, 4, and 7. What is the radius of the triangles inscribed circle?

r =

About 1.47.

Let s be half the perimeter of the triancle with sides a, b, c. Then the radius of the inscribed circle is given by:

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To find the radius ( r ) of the inscribed circle of a triangle with side lengths ( a ), ( b ), and ( c ), you can use the formula:

[ r = \frac{{\text{Area of the triangle}}}{{\text{Semiperimeter of the triangle}}} ]

where the semiperimeter ( s ) is calculated as:

[ s = \frac{{a + b + c}}{2} ]

and the area ( A ) of the triangle can be found using Heron's formula:

[ A = \sqrt{s(s - a)(s - b)(s - c)} ]

Substituting the given side lengths into these formulas:

[ s = \frac{{8 + 4 + 7}}{2} = \frac{19}{2} ]

[ A = \sqrt{\frac{19}{2} \left(\frac{19}{2} - 8\right)\left(\frac{19}{2} - 4\right)\left(\frac{19}{2} - 7\right)} ]

[ r = \frac{\sqrt{\frac{19}{2} \left(\frac{19}{2} - 8\right)\left(\frac{19}{2} - 4\right)\left(\frac{19}{2} - 7\right)}}{\frac{19}{2}} ]

[ r \approx \frac{\sqrt{\frac{19}{2} \times \frac{3}{2} \times \frac{15}{2} \times \frac{5}{2}}}{\frac{19}{2}} ]

[ r \approx \frac{\sqrt{\frac{19 \times 3 \times 15 \times 5}{16}}}{\frac{19}{2}} ]

[ r \approx \frac{\sqrt{\frac{21375}{16}}}{\frac{19}{2}} ]

[ r \approx \frac{\sqrt{21375}}{8.5} ]

[ r \approx \frac{5\sqrt{21375}}{17} ]

[ r \approx 5.081 ]

So, the radius of the triangle's inscribed circle is approximately ( 5.081 ) units.

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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.

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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