Problem Set for Previewing Unit 3
In the third unit of Math I students are beginning their understanding of proofs, properties of triangles, and properties of quadrilaterals. In this preview we will focus on some of the properties of triangles and some of the points of concurrency that lie within. The students need to be able to identify perpendicular bisectors, medians, angle bisectors, and altitudes on a triangle; in addition, the points of concurrency that occur within the triangle when looking at these 4 different types of lines. We do not want to limit the students understanding of a point of concurrency to only these four; however, these are the only four which students will see in Math I.
Here are some fun activities that can help students better understand the term center of mass.
From the Project Intermath Webpage, we are going to be using the Balancing the Triangular Teeter activity. Working with the class we will learn about the 4 points of congruency created by the perpendicular bisector, median, angle bisectors, and altitudes. To better introduce the where these points intersect we can use a paper folding activity found here (starts on page 7). Here is how to fold the paper to create the circumcenter, incenter and centriod. In order to explore the perpendicular bisector, median, angle bisector, and altitudes points of concurrency we can use this. This Geogebra file can help with showing students where the centers move depending on whether we are given an acute, right, or obtuse triangle.
Properties of Points of Concurrency
1) The circumcenter is equidistant to the vertices of the triangle
2) The circumcenter is the center point of a circle that intersects the three vertices of the triangle. This is called circumscribing a triangle.
1) The incenter is equidistant to the sides of the triangle
2) The incenter is the center of a circle that intersects the three sides of the triangle only once. This is called inscribing the circle.
1) The centriod is also known as the center of balance for a triangle.
2) The distance from the vertex to the centriod is 2 times the distance from the centriod to the side of the triangle