Chapter 5 Flashcards
What is the midsegment of a triangle?
a segment connecting the midpoint of two sides
Theorem 5-1 (midsegment theorem)
If a segment joins the midpoint of two sides of a triangle, then it is parallel to the third side and half it’s length
Perpendicular bisector Theorem
If a point is on the perpendicular bisector of a triangle, then it is equidistant from the endpoints of a segment.
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle
Converse of the Angle Bisector Theorem
If a point is in the interior of an angle is equidistant from the sides of angle, then the point is on the angle bisector
What does it mean if something is CONCURRENT?
3 or more lines intersect in one point
What is the POINT OF CONCURRENCY?
The point in which concurrent lines intersect
What are the four points of concurrency?
- Circumcenter
- Incenter
- Centroid
- Orthocenter
What is a CIRCUMCENTER?
where perpendicular bisectors intersect
What do you need to find to find the CIRCUMCENTER?
you must find the MIDPOINTS and PERPENDICULAR SLOPES
Theorem 5-6
this theorem talks about circumcenters
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. The point of concurrency is called the circumcenter.
What is an INCENTER?
where angle bisectors intersect
Theorem 5-7
(this theorem is for incenters)
The bisector of the angles of a triangle are concurrent at a point equidistant from the sides. The point of concurrency is called the incenter of the triangle.
What is a MEDIAN?
the segment of a triangle whose endpoints are a vertex and the midpoint of the opposite side
What is a CENTROID?
where the medians intersect
What do you need to find the CENTROID?
The midpoint and the vertices
Theorem 5-8
theorem for centroid
The medians of a triangle are concurrent at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side. The point of concurrency is called the centroid.
What is an altitude?
the perpendicular segment from a vertex to the line containing the opposite side (height)
What is an orthocenter?
where the altitudes intersect
What do you need to find the orthocenter?
the perpendicular slope and the vertex
Theorem 5-9
(this theorem is about the orthocenter)
The lines that contain the altitude of a triangle are concurrent. The point of concurrency is called the orthocenter of the triangle.
Theorem 5-10 (sides)
if two sides of a triangle are not congruent, then the larger angle lies opposite the larger side
Theorem 5-11 (angles and sides)
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle
in other words…the largest side is opposite the largest angle the middle angle is opposite the middle angle, and the shortest side is opposite the smallest angle
Theorem 5-12 (side lengths sum)
The sum of the lengths of any two sides of a triangle is greater than the length of the third side
OR
the sum of the 2 shortest sides must be greater than the third side
formula for side lengths
v-w < side < v+w
y < x < z
