A triangle has corners at #(3 , 3 )#, #(1 ,2 )#, and #(5 ,4 )#. What is the radius of the triangle's inscribed circle?

Question

A triangle has corners at #(3  ,  3    )#, #(1   ,2  )#, and #(5 ,4   )#.  What is the radius of the triangle's inscribed circle?

Savannah Aguilar · Accepted Answer

The three points lie on same straight line and hence no distinct triangle can be formed. In other words, one could say that radius of inscribed circle is #0#.

If the sides of a triangle are #a#, #b# and #c#, then the area of the triangle #Delta# is given by the formula #Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)# and radius of inscribed circle is #Delta/s# Hence let us find the sides of triangle formed by #(3,3)#, #(1,2)# and #(5,4)#. This will be surely distance between pair of points, which is #a=sqrt((1-3)^2+(2-3)^2)=sqrt(4+1)=sqrt5=2.236# #b=sqrt((5-1)^2+(4-2)^2)=sqrt(16+4)=sqrt20=4.472# and #c=sqrt((5-3)^2+(4-3)^2)=sqrt(4+1)=sqrt5=2.236# As #a+c=b#, it is apparent that the three points lie on same straight line and hence no distinct triangle can be formed. In other words, one could say that radius of inscribed circle is #0#.

Sadie Adams · Answer

undefined To find the radius of the inscribed circle in the given triangle, we can use the formula:

$$ r = \frac{A}{s} $$

Where:
- $ r $ is the radius of the inscribed circle.
- $ A $ is the area of the triangle.
- $ s $ is the semi-perimeter of the triangle.

First, we need to calculate the lengths of the sides of the triangle using the distance formula:

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

For the sides of the triangle, we have:
- Side 1: Between (3, 3) and (1, 2).
- Side 2: Between (1, 2) and (5, 4).
- Side 3: Between (5, 4) and (3, 3).

After calculating the lengths of the sides, we can find the semi-perimeter ($ s $) by summing up the lengths of the three sides and dividing by 2.

Then, we'll use Heron's formula to find the area ($ A $) of the triangle:

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

Where $ a $, $ b $, and $ c $ are the lengths of the three sides.

Once we have the area, we can find the radius ($ r $) using the formula mentioned earlier.

Let's perform these calculations:

Side 1: $\sqrt{(1 - 3)^2 + (2 - 3)^2} = \sqrt{4 + 1} = \sqrt{5}$

Side 2: $\sqrt{(5 - 1)^2 + (4 - 2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$

Side 3: $\sqrt{(5 - 3)^2 + (4 - 3)^2} = \sqrt{4 + 1} = \sqrt{5}$

Semi-perimeter: $s = \frac{\sqrt{5} + 2\sqrt{5} + \sqrt{5}}{2} = \frac{4\sqrt{5}}{2} = 2\sqrt{5}$

Area: $A = \sqrt{2\sqrt{5}(2\sqrt{5}-\sqrt{5})(2\sqrt{5}-\sqrt{5})(2\sqrt{5}+\sqrt{5})} = \sqrt{2\sqrt{5} \cdot \sqrt{5} \cdot \sqrt{5} \cdot 3\sqrt{5}} = \sqrt{2 \cdot 5 \cdot 5 \cdot 3} = \sqrt{150} = 5\sqrt{6}$

Now, using the formula $r = \frac{A}{s}$:

$r = \frac{5\sqrt{6}}{2\sqrt{5}} = \frac{5\sqrt{6}}{2}$