# A triangle has corners at #(7 , 9 )#, #(3 ,7 )#, and #(4 ,8 )#. What is the radius of the triangle's inscribed circle?

Incenter radius

Given the coordinates of the three vertices of a triangle ABC,

the coordinates of the incenter O are

Given A(7,9), B(3,7), C(4,8)

Using distance formula we can calculate a, b, c.

Semi perimeter

Area of triangle

Incenter radius

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To find the radius of the inscribed circle in a triangle, use the formula:

[ r = \frac{A}{s} ]

Where ( A ) is the area of the triangle and ( s ) is the semi-perimeter of the triangle.

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

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

Given the coordinates:

( (x_1, y_1) = (7, 9) )

( (x_2, y_2) = (3, 7) )

( (x_3, y_3) = (4, 8) )

[ a = \sqrt{(x_2 - x_3)^2 + (y_2 - y_3)^2} ] [ b = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} ] [ c = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} ]

Substitute the coordinates into the formulas and calculate:

[ A = \frac{1}{2} |7(7 - 8) + 3(8 - 9) + 4(9 - 7)| = \frac{1}{2} |7 - 3 + 8| = \frac{1}{2} \times 12 = 6 ]

[ a = \sqrt{(3 - 4)^2 + (7 - 8)^2} = \sqrt{1 + 1} = \sqrt{2} ] [ b = \sqrt{(4 - 7)^2 + (8 - 9)^2} = \sqrt{9 + 1} = \sqrt{10} ] [ c = \sqrt{(7 - 3)^2 + (9 - 7)^2} = \sqrt{16 + 4} = \sqrt{20} ]

[ s = \frac{\sqrt{2} + \sqrt{10} + \sqrt{20}}{2} ]

Now, plug the values of ( A ) and ( s ) into the formula for the radius of the inscribed circle:

[ r = \frac{6}{\frac{\sqrt{2} + \sqrt{10} + \sqrt{20}}{2}} ]

[ r = \frac{12}{\sqrt{2} + \sqrt{10} + \sqrt{20}} ]

[ r ≈ \frac{12}{1.41 + 3.16 + 4.47} ]

[ r ≈ \frac{12}{9.04} ]

[ r ≈ 1.33 ]

So, the radius of the triangle's inscribed circle is approximately ( 1.33 ) 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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