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 _________________.

parallel congruent not related concurrent

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

C-10 Perpendicular Bisector Concurrency Conjecture

The three perpendicular bisectors of a triangle are ___________________.

parallel congruent not related concurrent

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 ____________________.

concurrent not related parallel congruent

3.8 Investigation 2: Circumcenter- use the perpendicular bisector tool to find the circumcenter, and label the point X. Use the distance tool to find the distances AX, BX, and CX. Use your observations to answer the questions below.

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

C-12 Circumcenter Conjecture

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

equidistant, sides equidistant, vertices twice the distance, vertices as from the sides

null

3.8 Investigation 2: Incenter- Use the angle bisector tool to construct the incenter for the triangle, and label the point Y. Use the distance tool to find the distances from Y to each side of the triangle. Use your observations to answer the questions below.

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.

twice the distance, vertices as from the sides equidistant, vertices equidistant, sides

3.9 Investigations

3.9 Investigation 1- Use the midpoint tool to find the midpoint of each side and the segment tool to construct all 3 medians for the triangle. Use your observations to complete the conjecture below.

C-14 Median Concurrency Conjecture

The three medians of a triangle are _____________.

concurrent congruent not related parallel

3.9 Investigation 2- Centroid: Use the midpoint and segment tools to construct the centroid for the triangle. Label the midpoints as follows: P is the midpoint of AB, Q is the midpoint of BC, and R is the midpoint of AC. Label the centroid Z. Use the distance tool to measure and compare the following distances: CZ & ZP, BZ & ZR, and AZ & ZQ.

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.

twice the the same half the