Question

1. What are the properties of the circumcenter of a triangle?
2. What are the properties of the incenter of a triangle?
3. The lengths of three sides of a triangle are: n + 3, n + 6, and 3n – 6. Find the range of the possible lengths for n.

Answer 1

1. The circumcenter is the center of the circle through the triangles three vertices.
2. The incenter is the center of the largest circle that fits in the circle.
3. The lengths of each two sides of a trianlge must be greater thna the length of the third side.
The first of these inequalities (using the first two side lengths added are less than the third side length) is
(n + 3) + (n + 6) > 3n - 6
n + 3 + n + 6 > 3n - 6
2n + 9 > 3n - 6
-n > -15
n < 15
Now write an inequality for the first and third sides are greater than the second, and another for the second and third sides added are greater than the first one.