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 completely. the area of the triangle is a = \boxed{}

Explanation:

Step1: Split triangle by incenter

The incenter divides the triangle into 3 smaller triangles with height \(a\) and bases \(x,y,z\).

Step2: Calculate area of small triangles

Area of each small triangle: \(\frac{1}{2}xa\), \(\frac{1}{2}ya\), \(\frac{1}{2}za\).

Step3: Sum areas using distributive property

Total area \(A = \frac{1}{2}xa + \frac{1}{2}ya + \frac{1}{2}za = \frac{1}{2}a(x + y + z)\).

Answer:

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