A triangle has sides with lengths of 8, 4, and 6. What is the radius of the triangle's inscribed circle?

Answer 1

We can use semiperimeter and Heron's formula

First, we find the semiperimeter, which is the sum of the sides divided by two.

Therefore, #(8+4+6)/2 = 9# which is the semiperimeter
Now, we can use Heron's formula to find the area, which is #sqrt((s)(s-a)(s-b)(s-c)) = A#
We just plug in values now #sqrt((9)(9-8)(9-4)(9-6))#
which is #sqrt((9)(1)(5)(3))# which is #sqrt135#
Therefore, #sqrt135 = A#

Now, we can use the formula relating inradii, Area, and semiperimeter

#A = r * s#
Since we found two values, we can just plug in to find the third. #sqrt135 = r * 9#
#sqrt(135) / 9 = r#

And we are done.

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Answer 2

The radius of the triangle's inscribed circle can be found using the formula ( r = \frac{A}{s} ), where ( A ) is the area of the triangle and ( s ) is the semi-perimeter of the triangle. The area ( A ) can be calculated using Heron's formula:

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

where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle, and ( s ) is the semi-perimeter calculated as ( s = \frac{a + b + c}{2} ).

For the given triangle with side lengths 8, 4, and 6, the semi-perimeter ( s ) is ( s = \frac{8 + 4 + 6}{2} = 9 ). Using Heron's formula, we can find the area ( A ):

[ A = \sqrt{9(9-8)(9-4)(9-6)} = \sqrt{9 \cdot 1 \cdot 5 \cdot 3} = \sqrt{135} \approx 11.62 ]

Finally, we can calculate the radius of the inscribed circle ( r ):

[ r = \frac{A}{s} = \frac{11.62}{9} \approx 1.29 ]

So, the radius of the triangle's inscribed circle is approximately 1.29.

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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.

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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