Think about the triangle's perimeter or area. How can you involve x in these concepts?
x=AB+AC-BC/2 or x=AB* AC/AB+AC+BC
Practice makes perfect
To find an expression for x, we can use the fact that the area of â–ł ABC is equal to the area of the three smaller triangles, â–ł ABD, â–ł ADC, and â–ł BDC that's inside of â–ł 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
Since â–ł ABC is a right triangle, the base and height will be the length of its legs, AC and AB. For â–ł ABD, â–ł ADC, and â–ł BDC, the height is the perpendicular segment from the incenter. According to the Incenter Theorem, the incenter of a triangle is equidistant from the sides of the triangle. Therefore, the height of all of these triangles are x.
Now we can determine expressions for each triangle's area.
Triangle
b
h
1/2bh
Area
â–ł ABC
AB
AC
1/2(AB)(AC)
AB* AC/2
â–ł ABD
AB
x
1/2(AB)(x)
AB* x/2
â–ł ADC
AC
x
1/2(AC)(x)
AC* x/2
â–ł BDC
BC
x
1/2(BC)(x)
BC* x/2
When we have expressions for the area of the four triangles, we can equate the expression for the larger area with the sum of the three smaller triangles.
We can also write an expression for x by using the perimeter of the triangle. We have already established that the incenter, D, is equidistant, x, to each side. From the diagram, we can also identify a square with a side of x since it's a quadrilateral with four right angles and two congruent adjacent sides.
Also, since D is the incenter of â–ł ABC, it means that BD and CD are angle bisectors. This gives us enough information to show congruence between two pairs of triangles using the AAS Congruence Theorem.
Let's identify the remaining corresponding sides in each pair of congruent triangles and label them.
Examining the diagram, we see that the sum of AB and AC is 2x greater than BC. Therefore, we can write the following equation.
AB+AC-BC=2x
Now we can solve for x.
