A triangle has corners at #(2 ,3 )#, #(1 ,9 )#, and #(6 ,8 )#. What is the radius of the triangle's inscribed circle?

Answer 1

#(x-3.0195)²+(y-6.9143)²=1.6491^2#

The triangle vertices #p_1=(2,3),p_2=(1,9),p_3=(6,8)# First step is bissectrice building. #b_1->p_1+v_1 lambda_1# #b_2->p_2+v_2 lambda_2#
where: #v_1 = (p_2-p_1)/norm(p_2-p_1)+(p_3-p_1)/norm(p_3-p_1) = (0.460296, 1.76726)# #v_2 = (p_1-p_2)/norm(p_1-p_2)+(p_3-p_2)/norm(p_3-p_2)=(1.14498, -1.18251)#
The bissectrices intersection point is the circumference center calculated as the solution #(lambda_1^0,lambda_2^0)# to the system. #p_1+v_1 lambda_1 = p_2+v_2 lambda_2# The circumference center is obtained as #c=p_1+v_1 lambda_1^0= p_2+v_2 lambda_2^0 = (3.01951, 6.9143)# The circumference radius is obtained using Pythagoras. #r = sqrt(norm(c-p_1)^2-norm((c-p_1). (p_2-p_1)/norm(p_2-p_1))^2) = 1.64914#
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Answer 2

To find the radius of the triangle's inscribed circle, we can use the formula:

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

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

  1. First, we need to find the lengths of the sides of the triangle using the distance formula:

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

  1. Then, calculate the semi-perimeter ( s ) using the formula:

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

Where ( a ), ( b ), and ( c ) are the lengths of the sides.

  1. Next, calculate the area ( A ) of the triangle using Heron's formula:

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

  1. Finally, plug the values of ( A ) and ( s ) into the formula for the radius ( r ).

  2. The radius ( r ) is the desired value.

Let's proceed with these calculations.

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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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