Special Segments in Triangles

Interactive Drawings to Use

Perpendicular Bisectors in Triangles

• Recall that the perpendicular bisectors are concurrent at a point called the CIRCUMCENTER.
• The circumcenter is the center of a circle that can be circumscribed around the triangle.
• The circumcenter is equidistant from the vertices of the triangle.
• The distance from the circumcenter to each of the vertices is a radius of the circle.
• The circumcenter can lie inside, on, or outside of a triangle. (See following slides)

In an ACUTE triangle, the CIRCUMCENTER lies INSIDE the triangle.

In a RIGHT triangle, the CIRCUMCENTER lies ON the triangle at the midpoint of the hypotenuse.

Notice that the hypotenuse, BC, of the right triangle is a DIAMETER of the circumscribed circle.

In an OBTUSE triangle, the CIRCUMCENTER lies OUTSIDE the triangle.

Notice that the obtuse triangle appears to "fit" into less than half of the circle.

Angle Bisectors in Triangles

• Recall that the angle bisectors are concurrent at a point called the INCENTER.
• The incenter is the center of a circle that can be inscribed inside the triangle.
• The incenter is equidistant from the sides of the triangle.
• The distance from the incenter to each of the sides of the triangle is a radius of the inscribed circle.
• The incenter always lies inside the triangle, because the inscribed circle lies entirely inside the triangle.

The INCENTER lies inside an acute triangle.

The INCENTER lies inside a right triangle.

The INCENTER lies inside a right triangle.

Altitudes in Triangles

• An altitude is a segment from a vertex that is perpendicular to the opposite side (or to the line containing it).
• Altitude is the same thing as the height of the triangle.
• Recall the altitudes are concurrent in a point called the ORTHOCENTER.
• The orthocenter can lie inside, on, or outside of a triangle. (See following slides)

The ORTHOCENTER of an ACUTE triangle lies INSIDE the triangle.

The ORTHOCENTER of a RIGHT triangle lies ON the triangle.

Notice that the ORTHOCENTER of a RIGHT triangle lies ON the triangle at the VERTEX of the right angle, point C.

The ORTHOCENTER of an OBTUSE triangle lies OUTSIDE the triangle.

Notice that the sides that form the obtuse angle must be extended in order to make those two altitudes.

Notice that the orthocenter would lie "behind" point C, and would be formed where the extensions of the altitudes intersect.

Medians in Triangles

• A median is a segment that has as its endpoints a vertex of the triangle, and the midpoint of the opposite side.
• Recall the medians are concurrent at a point called the CENTROID.
• The centroid is the "balancing point," or center of mass, of a triangle.
• Continued on next slide...

Medians in Triangles

• The centroid divides each median into a 2:1 ratio. (It is closer to the midpoint than it is to the opposite vertex.)
• Or, the centroid lies on each median 2/3rds of the way from the vertex to the midpoint of the opposite side.

The centroid of an ACUTE triangle lies INSIDE the triangle.

The centroid of a RIGHT triangle lies INSIDE the triangle.

The centroid of an OBTUSE triangle lies INSIDE the triangle.

Untitled Slide