Question

a triangle with incenter p has side lengths x, y, and z. the distance from p to each side is a. write an expression for the area of the triangle. use the distributive property to factor your expression (factor completely.)
the area of the triangle is a = \square.

Explanation:

Step1: Recall triangle area with inradius

The area of a triangle can be expressed as the sum of the areas of three smaller triangles formed by the incenter and the sides. The area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. For each side (base) $x$, $y$, $z$ and height (distance from incenter) $a$, the areas of the three sub - triangles are $\frac{1}{2}xa$, $\frac{1}{2}ya$, and $\frac{1}{2}za$ respectively.

Step2: Sum the areas of sub - triangles

The total area $A$ of the original triangle is the sum of the areas of these three sub - triangles. So, $A=\frac{1}{2}xa+\frac{1}{2}ya+\frac{1}{2}za$.

Step3: Apply the distributive property

Using the distributive property $ab + ac+ad=a(b + c + d)$ (in reverse, here $a=\frac{1}{2}a$ and $b=x$, $c = y$, $d = z$), we can factor out $\frac{1}{2}a$ from the expression. So, $A=\frac{1}{2}a(x + y + z)$.

Answer:

$\frac{1}{2}a(x + y + z)$