• ### Unit 4: Triangle Relationships

In this unit, we will explore triangulation, midsegments, and bisected lengths of triangles. City planners and building designers regularly need to calculate the circumcenter of a triangle, which is the point in the middle of the triangle that is equally distant from all three sides. For example, they may need to calculate whether the location of a new parking garage is the same distance from three or more companies, or whether a city monument is exactly in the middle of a plaza. Construction workers use the midsegment of a triangle to help strengthen roof trusses.

Completing this unit should take you approximately 3 hours.

• ### 4.1: Midsegment Theorem

The first property of a triangle we explore is the midsegment theorem. A midsegment is a line that connects two midpoints of the sides of a triangle. The midsegment theorem states that a midsegment must be parallel to the third side of the triangle, and the midsegment must be half the length of the third side of the triangle.

• ### 4.2: Perpendicular Bisectors

Now, let's review perpendicular bisectors. A bisector is a line that cuts through the midpoint of a line segment. A perpendicular bisector is a line that bisects a segment and is perpendicular to that segment. Perpendicular bisectors have a special use in creating triangles.

• ### 4.3: Angle Bisectors

An angle bisector cuts angles exactly in half. We can use an angle bisector to create two identical triangles within the original angle.

• ### 4.4: Medians

The next property of triangles we explore is the median of a triangle. The median of a triangle is a segment that joins the vertex of the triangle to the midpoint of the opposite side.

• ### 4.5: Altitudes

The last important property of triangles we explore is the altitude of a triangle. The altitude of a triangle is a line segment that runs perpendicular from the vertex of the triangle to the opposite side of the triangle. Depending on the type of triangle (acute, right, or obtuse), the altitude may exist outside of the triangle, on a side of the triangle, or inside the triangle.

• ### 4.6: Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of two sides of a triangle must be greater than the length of the third side of the triangle. This allows us to determine if a triangle can be formed from objects of three given lengths.