# 3.8 and 3.9 Points of Concurrency Investigation

## 3.8 Investigation 1- Angle Bisectors: Use the angle bisector tool to construct all 3 angle bisectors for the triangle. Use your observations to complete the conjecture below.

## C-9 Angle Bisector Concurrency Conjecture

The three angle bisectors of a triangle are _________________.

## C-10 Perpendicular Bisector Concurrency Conjecture

The three perpendicular bisectors of a triangle are ___________________.

## 3.8 Investigation 1- Altitudes: use the perpendicular line tool to construct all 3 altitudes for the triangle. Use your observations to complete the conjecture below.

## C-11 Altitude Concurrency Conjecture

The 3 altitudes (or the lines containing the altitudes) of a triangle are ____________________.

How are the measures AX, BX, and CX related?

## C-12 Circumcenter Conjecture

The circumcenter of a triangle is _____________ from the ____________ of the triangle.

## null

How are the distances from Y to each of the sides related?

## C-13 Incenter Conjecture

The incenter of a triangle is _______________________ from the ______________ of the triangle.

## 3.9 Investigations

## C-14 Median Concurrency Conjecture

The three medians of a triangle are _____________.

## AQ is one median of the triangle.

How does point Z split AQ? In other words, how does AZ compare to ZQ?

## BR is one median of the triangle.

How does point Z split BR? In other words, how does BZ compare to ZR?

## CP is one median of the triangle.

How does Z split CP? In other words, how does CZ compare to ZP?

## C-15 Centroid Conjecture

The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is __________ distance from the centroid to the midpoint of the opposite side.