Transcript Lesson 1 Contents - Headlee's Math Mansion

Lesson 5-1
Bisectors, Medians and
Altitudes
Transparency 5-1
5-Minute Check on Chapter 4
Refer to the figure.
1. Classify the triangle as scalene, isosceles, or equilateral.
2. Find x if mA = 10x + 15, mB = 8x – 18, and
mC = 12x + 3.
3. Name the corresponding congruent angles if
RST  UVW.
4. Name the corresponding congruent sides if LMN  OPQ.
5. Find y if DEF is an equilateral triangle and mF = 8y + 4.
6.
Standardized Test Practice:
What is the slope of a line that contains
(–2, 5) and (1, 3)?
A
–2/3
B
2/3
C
–3/2
D
3/2
Transparency 5-1
5-Minute Check on Chapter 4
Refer to the figure.
1. Classify the triangle as scalene, isosceles, or equilateral.
isosceles
2. Find x if mA = 10x + 15, mB = 8x – 18, and
mC = 12x + 3.
6
3. Name the corresponding congruent angles if
RST  UVW.
R  U; S  V; T  W
4. Name the corresponding congruent sides if LMN  OPQ.
LM  OP; MN  PQ; LN  OQ
5. Find y if DEF is an equilateral triangle and mF = 8y + 4.
6.
Standardized Test Practice:
What is the slope of a line that contains
(–2, 5) and (1, 3)?
A
–2/3
B
2/3
7
C
–3/2
D
3/2
Objectives
• Identify and use perpendicular bisectors and
angle bisectors in triangles
• Identify and use medians and altitudes in
triangles
Vocabulary
• Concurrent lines – three or more lines that intersect at
a common point
• Point of concurrency – the intersection point of three or
more lines
• Perpendicular bisector – passes through the midpoint
of the segment (triangle side) and is perpendicular to
the segment
• Median – segment whose endpoints are a vertex of a
triangle and the midpoint of the side opposite the vertex
• Altitude – a segment from a vertex to the line
containing the opposite side and perpendicular to the
line containing that side
Vocabulary
• Circumcenter – the point of concurrency of the
perpendicular bisectors of a triangle; the center of the
largest circle that contains the triangle’s vertices
• Centroid – the point of concurrency for the medians of a
triangle; point of balance for any triangle
• Incenter – the point of concurrency for the angle
bisectors of a triangle; center of the largest circle that
can be drawn inside the triangle
• Orthocenter – intersection point of the altitudes of a
triangle; no special significance
Theorems
• 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
the segments lies on the perpendicular bisector of a
segment.
• Theorem 5.3, Circumcenter Theorem – The circumcenter of
a triangle is equidistant from the vertices of the triangle.
• Theorem 5.4 – Any point on the angle bisector is equidistant
from the sides of the triangle.
• Theorem 5.5 – Any point equidistant from the sides of an
angle lies on the angle bisector.
• Theorem 5.6, Incenter Theorem – The incenter of a triangle
is equidistant from each side of the triangle.
• Theorem 5.7, Centroid Theorem – 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.
Triangles – Perpendicular Bisectors
A
Note: from Circumcenter
Theorem: AP = BP = CP
Z
Midpoint
of AC
Circumcenter
Midpoint
of AB X
P
C
Y
Midpoint
of BC
B
Circumcenter is equidistant from the vertices
Triangles – Angle Bisectors
A
Note: from Incenter Theorem:
QX = QY = QZ
Z
Q
Incenter
C
X
Y
B
Incenter is equidistant from the sides
A
Triangles – Medians
Note: from Centroid theorem
BM = 2/3 BZ
Midpoint
Z of AC
Midpoint
of AB X
Centroid
M
C
Median
from B
Y Midpoint
of BC
B
Centroid is the point of balance in any triangle
A
Triangles – Altitudes
Note: Altitude is the shortest distance
from a vertex to the line opposite it
Z
Altitude
from B
C
X
Y
B
Orthocenter has no special significance for us
Special Segments in Triangles
Name
Type
Point of
Concurrency
Center Special
Quality
From
/ To
Equidistant
from vertices
None
midpoint of
segment
Incenter
Equidistant
from sides
Vertex
none
Vertex
midpoint of
segment
Perpendicular
Line,
Circumcenter
bisector
segment or
ray
Angle
bisector
Line,
segment or
ray
Median
segment
Centroid
Center of
Gravity
Altitude
segment
Orthocenter
none
Vertex
none
Location of Point of Concurrency
Name
Point of Concurrency
Perpendicular bisector
Circumcenter
Triangle Classification
Acute
Right
Obtuse
Inside hypotenuse Outside
Angle bisector
Incenter
Inside
Inside
Inside
Median
Centroid
Inside
Inside
Inside
Altitude
Orthocenter
Inside Vertex - 90 Outside
Given:
Find:
mDGE
Proof:
Statements
Reasons
1.
1. Given
2.
3.
4.
5.
2. Angle Sum Theorem
3. Substitution
4. Subtraction Property
5. Definition of angle
bisector
6. Angle Sum Theorem
7. Substitution
8. Subtraction Property
6.
7.
8.
Given:
.
Find: mADC
Proof:
Statements.
Reasons
1.
1. Given
2.
3.
4.
5.
2. Angle Sum Theorem
3. Substitution
4. Subtraction Property
5. Definition of angle
bisector
6. Angle Sum Theorem
7. Substitution
8. Subtraction Property
6.
7.
8.
ALGEBRA Points U, V, and W
are the midpoints of
respectively. Find a, b, and c.
Find a.
Segment Addition Postulate
Centroid Theorem
Substitution
Multiply each side by 3 and simplify.
Subtract 14.8 from each side.
Divide each side by 4.
Find b.
Segment Addition Postulate
Centroid Theorem
Substitution
Multiply each side by 3 and simplify.
Subtract 6b from each side.
Subtract 6 from each side.
Divide each side by 3.
Find c.
Segment Addition Postulate
Centroid Theorem
Substitution
Multiply each side by 3 and simplify.
Subtract 30.4 from each side.
Divide each side by 10.
Answer:
ALGEBRA Points T, H, and G are the midpoints of
respectively. Find w, x, and y.
Answer:
Summary & Homework
• Summary:
– Perpendicular bisectors, angle bisectors, medians
and altitudes of a triangle are all special segments
in triangles
– Perpendiculars and altitudes form right angles
– Perpendiculars and medians go to midpoints
– Angle bisector cuts angle in half
• Homework:
– Day 1: pg 245: 46-49
– Day 2: pg 245: 51-54