Geometry
Chapter 5: Relationships Within Triangles
Day
Topic
Name____________________________________
Period__________________________________
Assignment
Score
1
5.1 Midsegment theorem and coordinate proof.
What is a midsegment of a triangle? it is a segment that connects the midpoints of two sides of the triangle. Every
triangle has three midsegments.
Midsegment theorem of a triangle: The segment connecting the midpoints of two sides of a triangle is parallel to the
third side and is half as long as that side.
What is a coordinate proof? It involves placing geometric figures in a coordinate plane and using appropriate variables
to represent the coordinates of the figure.
5.1 Workbook.
1-21 all
Mixed review
p. 301 47-52
/4
2
5.2 Use perpendicular bisectors.
What is a perpendicular bisector? A perpendicular bisector intersects two lines, rays, or line segments at the midpoint
and creates two congruent segments as well as right angles.
What is meant by equidistant? A point is equidistant from two figures if the point is the same distance from each figure.
Perpendicular bisector theorem: In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant
from the endpoints of the segment.
Converse of perpendicular bisector theorem: In a plane, if a point is equidistant from the endpoints of a segment, then
it is on the perpendicular bisector of the segment.
What is concurrent? It is when three or more lines, rays, or segments intersect in the same point, they are called
concurrent lines, rays, or segments. The point where they intersect is called the point of concurrency.
Concurrency of perpendicular bisectors theorem: The perpendicular bisectors of a triangle intersect at a point that is
equidistant from the vertices of the triangle.
What is circumcenter? It is the point where the three perpendicular bisectors of a triangle intersect. The circumcenter is
the center of the circle and will be equidistant from the vertices of the triangle (think: the radius of the circle is equal to
all other radius of the circle).
Where is the location of the circumcenter in an acute, right or obtuse triangle?
Create a circle with center P that is circumscribed about a triangle, that is, a triangle inside a circle.
a) Acute triangle where P (the circumcenter, that is, the center) is inside the triangle.
b) Right triangle where P (the circumcenter, that is, the center) is on the triangle.
c) Obtuse triangle where P (the circumcenter, that is, the center) is outside the triangle.
5.2 Workbook.
1-24 all
Mixed review
p. 309 34-41
/4
3
5.3 Use angle bisectors of triangles.
What is an angle bisector? It is a ray that divides an angle into two congruent angles.
Angle bisector theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Converse of angle bisector theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle,
5.3 Workbook.
1-18 all
Mixed review
p. 316 39-47
/4
then it lies on the bisector of the angle.
Concurrency of angle bisectors theorem: the angle bisector of a triangle intersect at a point that is equidistant from the
sides of the triangle.
What is incenter? It is the point of concurrency of the three angle bisectors of a triangle; it always lies inside the
triangle.
Inscribed: if a circle is inscribed about a triangle it means that the circle is within the triangle.
4
5.4 Use medians and altitudes.
What is a median of a triangle? It is a segment from a vertex to the midpoint of the opposite side.
The three medians of a triangle are concurrent (they intersect at the same point).
What is a centroid? It is the point of concurrency, that is, where the three medians intersect. It is the point where the
triangle balances.
Concurrency of medians theorem: 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.
What is an altitude? It is the perpendicular segment from the vertex to the opposite side or to the line that contain the
opposite side. (Think: it is the height)
Concurrency of altitudes theorem: The lines containing the altitudes of a triangle are congruent.
Orthocenter: it is the intersection of the three altitudes of a triangle.
Where is the orthocenter found in an acute, right, or obtuse triangle?
5.4 Workbook.
1-26 all
Mixed review
p. 325 46-55
/4
5
5.5 Use inequalities in a triangle.
Describe the relationship between the angle and its opposite side. (hint: Shorter side means smaller angle)
If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle
opposite the shorter side.
If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side
opposite the smaller side.
Triangle inequality theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third
side.
5.5 Workbook.
1-32 all
Mixed review
p. 334 49-54
/4
6
Chapter 5 review.
Chapter 5 practice test.
pp. 344,347
1-24 all
p. 348 1-1 all
/4
7
Chapter 5 Exam. You have one shot at it so give it your very best effort.
/4
/24
I still have questions about: