How do you inscribe a circle in a triangle? Can you use areas to find the length of the circle's radius?
Angle bisectors. Radius: ≈ 2.83 in.
Practice makes perfect
Let's first draw the triangle.
The exercise has two parts, first we have to find the largest possible circle and second, we have to find the radius of this circle.
Finding the largest possible circle
To achieve the largest possible circle, we want to determine the triangle's incenter. Any circle with it's midpoint at the incenter of a circle will be inscribed in the circle which means it's as large it can be. To locate the incenter, we have to determine at least two angle bisectors for each of the triangle's vertices.
We will repeat the procedure for one more angle.
Now we have enough information to plot the incenter and then draw the inscribed circle.
Finding the radius
To find the radius of the circle, we need to add some information to the diagram. According to the Incenter Theorem, the incenter of a triangle is equidistant from the sides of the triangle. Let's add these segments, the length of which we will call r. We will also add a fourth segment from the vertex angle to the incenter.
To find an expression for r, we can use the fact that the area of △ ABC is equal to the area of the three smaller triangles, △ ABD, △ ADC, and △ BDC that's inside of △ ABC.
To determine the area of a triangle, we need it's base and height. Then, we can use the formula for calculating the area of a triangle.
Area=1/2bh
To determine the area of the blue triangle, we need it's height. Since it's an isosceles triangle, the angle bisector to the vertex angle will be a perpendicular bisector to AB and therefore the height of the triangle. This gives us enough information to calculate the height of △ ABC using the Distance Formula.
Now we have the height of △ ABC. Also, notice that the height of the smaller triangles that's inside of △ ABC are all r. Now we can form an equation that equates the area of △ ABC with the sum of the areas of △ ABD, △ ADC, and △ BDC.
1/2(sqrt(128))(8)= 1/2(8)(r)+ 1/2(12)(r)+1/2(12)(r)
Let's solve this equation.
