Question

since point d is the center of the circle that will be inscribed, and the inscribed circle will intersect ab at point e, ab is tangent to the inscribed circle. since de is perpendicular to tangent ab, de is the radius of the largest circle that will fit within triangle abc

part c
with point d as the center, create a circle passing through point e. measure the radius of the inscribed circle. would the radius be different if you used a line perpendicular to bc instead of ab to create the circle? explain your reasoning

Explanation:

The in - radius of a triangle is the same regardless of which side's perpendicular is considered for constructing the inscribed circle. This is because the center of the inscribed circle (in - center) is equidistant from all sides of the triangle. So, the radius of the circle created using a perpendicular to $\overline{BC}$ would be the same as the radius of the circle created using a perpendicular to $\overline{AB}$.

Answer:

The radius would not be different. The in - center of a triangle is equidistant from all sides of the triangle, so the radius of the inscribed circle (the distance from the in - center to any side) is constant.