Transcript Special Segments in Triangles

Special Segments in Triangles
Perpendicular bisector: A line or line
segment that passes through the midpoint
of a side of a triangle and is perpendicular
to that side.
Theorem 5-1-: Any point on the
perpendicular bisector of a segment is
equidistant from the endpoints of the
segment.
Theorem 5-2-: Any point equidistant from the
endpoints of a segment lies on the perpendicular
bisector of the segment.
Median and Altitude
Median: A segment that connects a
vertex of a triangle to the midpoint of
the side opposite to that vertex. Every
triangle has three medians.
Altitude: A segment that has an endpoint at
a vertex of a triangle and the other on the line
opposite to that vertex, so that the segment is
perpendicular to this line. Do example 1, page 239
Altitudes of
a right triangle
Altitudes of
an obtuse
triangle
Examples
1) ABC [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 ABC.
b) Determine if CP is an altitude of ABC
2) SGB [S(4,7), G(6,2), and B(12,-1)]:
a) Determine the coordinates of point J
on GB so that SJ is a median of SGB
b) Point M(8,3). Is GM an altitude of
SGB ?
Angle Bisector of a Triangle
Angle bisector of a triangle:
A segment that bisects an angle of
a triangle and has one endpoint at
a vertex of the triangle and the
other endpoint at another point on
the triangle.
Theorem 5-3: Any point on the bisector of an angle is
equidistant from the sides of the angle.
Theorem 5-4: 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.
Exaples to do
1. Do example 2, p. 240
2. Prove that if a triangle is equilateral, then an
angle bisector is also a median.
3. Do example 3, p. 241
Right Triangles - LL
A
IF AB  RS
BC  ST
<B  <S (both are right angles)
Then, ABC  RST : SAS
R
B
C
S
T
Theorem 5-5 : If the legs of one
right triangle are congruent to
the corresponding legs of another
right triangle, then the triangles
are congruent.(LL)
Right Triangles - HA
Theorem 5-6 : If the hypotenuse and
an acute angle of a right triangle are
congruent to the hypotenuse and
corresponding acute angle of another
right triangle, then the two triangles
are congruent.(HA) Prove Th.5-6
Right Triangles - LA
A
Complete the two-column proof (paper)
Case 2
Case 1
<A  <R
<C  <T
AB  RS
AB  RS
R
B
C
S
T
Theorem 5-7 : If one leg and an acute angle of one
right triangle are congruent to the corresponding leg
and acute angle of another right triangle, then the triangles are congruent.(LA) (Do example 2,p.247)
Right Triangles - HL
Postulate 5-1 : If the hypotenuse and theleg of a right
triangle are congruent to the hypotenuse and
corresponding leg of another right triangle, then the
two triangles are congruent.(HL) Example 3, p248
Indirect Proof (Indirect
reasoning)
Steps for writing and indirect proof:
1. Assume that the conclusion is false
2. Assume that the assumption leads to a
contradiction of the hypothesis or some other fact,
such as a postulate, theorem, corollary.
3. Point out that the assumption must be false and,
therefore, the conclusion must be true. (Ex.1p252)
State the assumption you would make to start an
indirect proof of each statement: AB bisects <A,
XTZ is isosceles, m<1 < m<2
INEQUALITY
Theorem 5-8 ( Exterior Angle Inequality Theorem):
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.
Definition of inequality: For any real numbers a and
b, a>b if and only if there is a positive number c such
that a = b + c
Inequalities for Sides and
Angles of a Triangle
Theorem 5-9: If one side of a triangle is longer than
another side, then the angle opposite to the longer
sidehas a greater measure than the angle opposite to
the shorter side
Theorem 5-10: If one angle of a triangle has a
greatermeasure than another angle, then the side
oppositeto the greater angle is longer than the side
opposite tothe lesser angle.
Cont...Inequalities for Sides and
Angles of a Triangle
Theorem 5-11: The perpendicular segment
from a point to a line is the shortest segment
from the point to the line.
Corollary 5.1: The perpendicular segment
from a point to a plane is the shortest
segment from the point to the plane.
The Triangle Inequality
Triangle Inequality Theorem: The sum of the lengths
of any two sides of a triangle is greater than the
length of the third side.
Ex. 1 p.267 Example 1: If 18, 45, 21 and 52 are
the lengths of segments, what is the probability
that a triangle can be formed if three of these
numbers are chosen at random as lengths of
the sides? (Ex.2 and 3-students do)
Inequalities Involving Two Triangles
SAS Inequality (Hinge Theorem):
If two sides of one triangle are congruent to two
sides of another triangle, and the included angle
in one triangle is greater than the included angle
in the other, then the third side of the first
triangle is longer than the third side in the
second triangle.
SSS Inequality (Theorem): If two sides of one
triangle are congruent to two sides of another
triangle, and the third side of the first triangle is
longer than the third side in the other, then the
angle between the pair of congruent sides in the
first triangle is greater than the corresponding
angle in the second triangle.