Goal: This lesson introduces the centroid of a triangle. By now, students should be familiar with the three main
intersection points regarding triangles: the circumcenter, the incenter, and the centroid.
History Connection!In addition to an infamous dictator, it appears Napoleon Bonaparte was an excellent mathemati-
cian. He was the top mathematics student in his school, taking algebra, trigonometry, and conics. His favorite class,
however, was geometry. After graduation, Bonaparte interviewed for a position in the Paris Military School and was
accepted due to his mathematical ability. Bonaparte completed the curriculum in one year (it took average students
two or three years to complete) and was appointed to the mathematics section of the French National Institute.
During his reign, Bonaparte appointed such men as Gaspard Monge, Joseph Fourier, and Pierre Laplace to recruit
teachers and reform the curriculum to emphasize mathematics. Napoleon’s Theorem is named as such because,
while Napoleon was not the first person to discover it, he supposedly found it independently.
Altitudes in Triangles
Pacing:This lesson should take one class period
Goal:Students will learn how to construct an altitude and how this auxiliary line differs from the median. Altitudes
are important in such geometrical concepts as area and volume.
Guided Discovery Questions!What is the difference between a median and an altitude? Is a median always an angle
bisector? Can the perpendicular bisector be a median?
Review Question:Using the diagram below, ask students to label the following auxiliary items: median, circumcen-
ter, incenter, orthocenter, altitude, perpendicular bisector
Inequalities in Triangles
Pacing:This lesson should take one class period
Goal: The purpose of this lesson is to familiarize students with the angle inequality theorems and the Triangle
Inequality Theorem. The lesson further extends the concepts of perpendicular lines and triangles to deduce the
shortest path between a point and a line is its perpendicular, thus leading to parallel lines.
If you have not used the in-class activity found in chapter 1,Classifying Triangleslesson, include it here. Otherwise,
reintroduce the concept in this lesson.
Additional Example:Jerry is across the street in the following diagram. Draw the path she should travel to minimize
the distance across the street.
1.5. Relationships Within Triangles