Chapter 5: Relationships in Triangles

5.1: Bisectors of Triangles

Vocabulary

A bisector splits an angle or segment into two congruent parts.
The angle bisector theorem states that it is an equal distance from the angle bisector to each side of the angle is equal if the line is perpendicular to the side.
The perpendicular bisector is the line that bisects a line and is perpendicular to that line.
The perpendicular bisectors theorem states that if any point on the perpendicular bisector is the same distance from the two base endpoints.

Angle Bisector Theorem

../_images/51-angle_bisector.svg

If \(\overrightarrow{AC}\) bisects \(\angle BAD\), then \(\overline{BC} \cong \overline{CD}\).

Perpendicular Bisectors

../_images/51-perpendicular_bisector.svg

\(\overleftrightarrow{AC}\) is the perpendicular bisector as \(\overline{AD} \cong \overline{AB}\). Point A is equal to the distance from point B than to point D. Since \(m \angle ACB = 90^\circ\), point C is the midpoint of \(\overline{DB}\).

5.2: Medians and Altitudes of Triangles

Vocabulary

A median of a triangle is a line segment originating at a vertex and ending at the midpoint of the oposite side.
An altitude of a triangle is a line segment originating at a vertex and perpendicular to the oposite side.
A angle bisector of a triangle is a ray that bisects a vertex.
A perpendicular bisector of a triangle is a line segment that forms a right angle on a side originating at the midpoint of that side.
The centroid of a triangle is where all three medians meet, the center of mass.

Example

../_images/52-vocabulary.svg

\(\overline{BE}\) is a median. \(\overline{DF}\) is a altitude. \(\overrightarrow{AC}\) is a angle bisector. \(overline{GC}\) is a perpendicular bisector.

How to find The Centroid of a Triangle (long way)?

../_images/52-centroid_finding.svg

  1. Find the midpoint of a side

\[\begin{split}midpoint_{BC} &= ( \frac{7 - 5}{2}, \frac{-3 - 3}{2} ) \\ midpoint_{BC} &= (1, -3) \\\end{split}\]

  1. Find the equation (use point slope)

\[\begin{split}y - 6 &= \frac{9}{2} (x - 3) \\ y - 6 &= \frac{9}{2} x - \frac{27}{2} \\\end{split}\]

  1. Solve using systems of equations (\(( \frac{5}{3}, 0)\) )

Note

No need to find the third median as it would go though the same point.

How to find The Centroid of a Triangle (short way)?

Find the average of all three x and y values.

\[\begin{split}centroid &= (\frac{3 - 5 + 7}{3}, \frac{6 - 3 - 3}{3}) \\ centroid &= (\frac{5}{3}, 0) \\\end{split}\]

5.3: Inequalities in One Triangle

5.4: Indirect Proof

5.5: The Triangle Inequality

5.6: Inequalities in Two Triangles

Last Minute Notes :)

The Centroid Theorem states that the medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. This point is called the centroid.
The Incenter Theorem states that the angle bisectors of a triangle intersect at a point equidistant from the sides of the triangle.

The point is called the incenter. | The Circumcenter Theorem states that the perpendicular bisectors of the sides of a triangle intersect at a single point. This point is equidistant from the vertices of the triangle and is called the circumcenter.