Final answer:
The area of a triangle with incenter P and sides of length x, y, and z can be calculated by adding the areas of three smaller triangles with base lengths x, y, and z, and all with the same height 'a'. This gives the expression for the area which can be factored using the distributive property to ½ × a × (x + y + z).
Step-by-step explanation:
The area of a triangle with incenter P can be calculated as the sum of the areas of three smaller triangles formed by drawing lines from P to the vertices. The area of each smaller triangle will be ½ × base × height, with 'a' being the height (distance from P to each side) and 'x', 'y', and 'z' being the bases. Therefore, the total area of the triangle can be expressed as:
Area = ½ × a× x + ½ × a × y + ½× a × z
To factor this expression using the distributive property:
Area = ½ × a ×(x + y + z)
The factor ½ × a represents the uniform distance from the incenter P to each side of the triangle.