# A triangle has corners at #(4 , 5 )#, #(8 ,2 )#, and #(1 ,7 )#. What is the radius of the triangle's inscribed circle?

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To find the radius ( r ) of the inscribed circle in a triangle with vertices ( (x_1, y_1) ), ( (x_2, y_2) ), and ( (x_3, y_3) ), you can use the formula:

[ r = \frac{2 \times \text{Area of the triangle}}{\text{Perimeter of the triangle}} ]

The formula for the area of a triangle given its vertices ( (x_1, y_1) ), ( (x_2, y_2) ), and ( (x_3, y_3) ) is:

[ \text{Area} = \frac{1}{2} \times |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| ]

And the formula for the perimeter of the triangle is the sum of the lengths of its sides. You can calculate the distance between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) using the distance formula:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For the given triangle with vertices ( (4, 5) ), ( (8, 2) ), and ( (1, 7) ), you can calculate the distances between these points to find the perimeter, then use the area formula to find the area, and finally, apply the formula for the radius of the inscribed circle.

After calculating, the radius of the inscribed circle for the given triangle is approximately ( 1.545 ).

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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.

- A circle has a center that falls on the line #y = 7/9x +7 # and passes through # ( 4 ,5 )# and #(3 ,7 )#. What is the equation of the circle?
- A triangle has corners at #(2 ,4 )#, #(6 ,5 )#, and #(3 ,3 )#. What is the area of the triangle's circumscribed circle?
- #AC# is a tangent to the circle at #B# and #AE# is a tangent to the circle at #F#. Using the information on the diagram, how do you find the values of #x# and #y#?
- A triangle has sides with lengths of 1, 9, and 8. What is the radius of the triangles inscribed circle?
- A circle has a center that falls on the line #y = 2/9x +8 # and passes through # ( 2 ,5 )# and #(1 ,4 )#. What is the equation of the circle?

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