Home > Geometry > Area of Inscribed Triangle

A triangle has corners at #(8 , 9 )#, #(7 ,3 )#, and #(2 ,4 )#. What is the radius of the triangle's inscribed circle?

Answer 1

Aurora Ayala

The radius of the incircle is #=1.63#

The area of the triangle is

#A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#

#A=1/2|(8,9,1),(7,3,1),(2,4,1)|#

#=1/2(8*|(3,1),(4,1)|-9*|(7,1),(2,1)|+1*|(7,3),(2,4)|)#

#=1/2(8(3-4)-9(7-2)+1(28-6))#

#=1/2(-8-45+22)#

#=1/2|-31|=31/2#

The length of the sides of the triangle are

#a=sqrt((8-7)^2+(9-3)^2)=sqrt(37)#

#b=sqrt((7-2)^2+(3-4)^2)=sqrt26#

#c=sqrt((8-2)^2+(9-4)^2)=sqrt61#

Let the radius of the incircle be #=r#

Then,

The area of the circle is

#A=1/2r(a+b+c)#

The radius of the incircle is

#r=(2a)/(a+b+c)#

#=(31)/(sqrt37+sqrt26+sqrt61)#

#=31/18.99=1.63#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Aurora Ayala

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

$r = \frac{2A}{P}$

where $r$ is the radius of the inscribed circle, $A$ is the area of the triangle, and $P$ is the perimeter of the triangle.

First, we need to find the area of the triangle using Heron's formula:

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

where $s$ is the semiperimeter of the triangle, and $a$ , $b$ , and $c$ are the lengths of the sides of the triangle.

$s = \frac{a + b + c}{2}$

Then, we can find the lengths of the sides of the triangle using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Finally, we can substitute the values into the formula for the radius of the inscribed circle to find the solution.

By signing up, you agree to our Terms of Service and Privacy Policy

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.