The three angle bisectors of a triangle are _________________.
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.
The three perpendicular bisectors of a triangle are ___________________.
The 3 altitudes (or the lines containing the altitudes) of a triangle are ____________________.
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?
The circumcenter of a triangle is _____________ from the ____________ of the triangle.
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?
The incenter of a triangle is _______________________ from the ______________ of the triangle.
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.
The three medians of a triangle are _____________.
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.
How does point Z split AQ? In other words, how does AZ compare to ZQ?
AZ is twice the length of ZQ
How does point Z split BR? In other words, how does BZ compare to ZR?
BZ is twice the length of ZR.
How does Z split CP? In other words, how does CZ compare to ZP?
CZ is twice the length of ZP.
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.