Download Geometry 5-1 Bisectors, Medians, and Altitudes

Geometry 5-1 Bisectors, Medians, and Altitudes
A. Special Segments
1. ________________________ __________________ -the perpendicular
bisector does what it sounds like, it is perpendicular to a segment and it bisects
the segment.
-DF is a perpendicular bisector of AB in
ABC
C
D
A
B
F
a.) Theorem 5-1 Any point on the perpendicular bisector of a segment is
equidistant from the endpoints of the segment.
b.) Theorem 5-2 Any point equidistant from the endpoints of a segment lies on
the perpendicular bisector of the segment.
c.) Since a triangle has three sides, there are three perpendicular bisectors in a
triangle. The perpendicular bisectors of a triangle intersect at a common point.
When three or more lines intersect at a common point, the lines are called
____________________ lines.
d.) Their point of intersection is called the _______________ of
__________________.
e.) The point of concurrency of the perpendicular bisectors of a triangle is called
the ___________.
f.) Theorem 5-3 Circumcenter Theorem -The circumcenter of a triangle is
equidistant from the vertices of the triangle.
2. ________________ _________________ -A segment that bisects an angle of
the triangle and has one endpoint at a vertex of the triangle and the other endpoint
at another point on the triangle.
C
A
R
B
a.) Theorem 5-4 -Any point on the bisector of an angle is _____________ from
the sides of the angle.
b.) Theorem 5-5 -Any point on or in the interior of an angle and equidistant from
the sides of an angle lies on the bisector of the angle.
c.) The three angle bisectors in any triangle are concurrent and meet at the
___________________ of the triangle.
d.) Theorem 5-6 Incenter Theorem -The incenter of a triangle is equidistant
from each side of the triangle
3. Median -A segment that connects the vertex of a triangle to the midpoint of the
side opposite that vertex.
C
A
B
F
a.) The medians of a triangle also intersect at a common point, the point of
_________________ for the medians of a triangle is called a centroid.
b.) Theorem 5-7 Centroid Theorem -The centroid of a triangle is located twothirds of the distance from a vertex to the midpoint of the side opposite the vertex
on a median.
4. Altitude -It has one endpoint at a vertex of a triangle and the other on the line
so that the segment is perpendicular to this line.
C
A
D
B
a.) The intersection point of the altitudes of a triangle is called the orthocenter.
Ex 1:
ABC has vertices at A(-3, 10), B(9, 2), and C(9, 15).
a.) Determine the coordinates of point P on AB so that CP is a median of
b.) Determine if CP is an altitude of
ABC
.
ABC.
Chapter 5: Relationships in Triangles
Section: 5.1 Bisectors, Medians, and Altitudes
Vocabulary:
_________________________________________: A line,
segment, or ray that bases through the midpoint of a side of a triangle
and is perpendicular to that side.
_________________________________________: The
intersection point of the three perpendicular bisectors of a triangle.
_________________________________________: A
segment whose endpoints are a vertex and the midpoint of the side
opposite the vertex.
_________________________________________: The
intersection point of the three medians of a triangle.
_________________________________________: A
segment from a vertex to the opposite side and perpendicular to it.
_________________________________________: The
intersection point of the three altitudes of a triangle.
Theorems:
5.1 Any point on the perpendicular bisector of a segment is
equidistant from the endpoints of the segment.
5.2 Any point equidistant from the endpoints of a segment lies
on the perpendicular bisector of the segment.
5.3 Circumcenter Theorem – The circumcenter of a triangle is
equidistant from the vertices of the triangle.
5.4 Any point on the angle bisector is equidistant from the
sides of the angle.
5.5 Any point equidistant from the sides of an angle lies on the
angle bisector.
5.6 ________________________________________ – The
incenter of a triangle is equidistant from each side of the triangle
5.7 ________________________________________ – The
centroid of a triangle is located two thirds of the distance from a
vertex to the midpoint of the side opposite the vertex on a median.
Section 5.2: Inequalities and Triangles
Properties of inequalities for Real Numbers
For all numbers a, b, and c
a < b, a = b, or a > b
ttttttttttttttttttttttttttttttttttttttt
1) If a < b and b < c, then a < c.
2) If a > b and b > c, then a > c.
1) If a > b and a + c > b + c,
and a – c > b – c.
2) If a < b and a + c < b + c,
and a – c < b – c.
1) If c > 0 and a < b,
then ac < bc and a/c < b/c.
2) If c > 0 and a > b,
then ac > bc and a/c >b/c.
3) If c < 0 and a < b,
then ac > bc and a/c > b/c.
4) If c < 0 and a > b,
then ac < bc and a/c < b/c.
Theorems:
5.8 _______________________________________________
– If an angle is an exterior angle of a triangle , then its measure is
greater than the measure of either of its corresponding remote interior
angles.
5.9 If one side of a triangle is longer than another side, then the
angle opposite the longer side has a greater measure than the angle
opposite the shorter side.
5.10 If one angle of a triangle has a greater measure than
another angle, then the side opposite the greater angle is longer than
the side opposite the shorter angle.
Section 5.3: Indirect Proof
Steps for writing an Indirect Proof (Proof by Contradiction)
1)
Assume that the conclusion is false.
2)
Show that this assumption leads to a contradiction of
the hypothesis, or some other fact, such as a definition,
postulate, or corollary.
3)
Point out that because the false conclusion leads to an
incorrect statement, the original conclusion must be
true.
Section 5.4: The Triangle Inequality
Theorems:
5.11 _______________________________________________
– The sum of the lengths of any two angle is greater than the length
of the third side.
5.12 The perpendicular segment from a point to a line is the
shortest segment form the point to the line.
Corollaries:
5.1 The perpendicular segment from a point to a plane is the
shortest segment from the point to the plane.
Section 5.5: Inequalities Involving Two Triangles
Theorems:
5.13 ______________________________________________–
If two sides of a triangle are congruent to two sides of another
triangle and the included angle in one triangle is has a greater
measure than the included angle in the other triangle, then the third
side of the first triangle is longer than the third side of the second
triangle.
5.14 ______________________________________________–
If two sides of a triangle are congruent to two sides of another
triangle and the third side in one triangle is has a greater measure than
the third side in the other triangle, then the angle between the pair of
congruent sides in the first triangle is greater than the angle between
the pair of congruent sides in the second triangle.