5.1 Perpendiculars and Bisectors Day 1 Part 1 CA Standard 16.0.
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Transcript of 5.1 Perpendiculars and Bisectors Day 1 Part 1 CA Standard 16.0.
5.1 Perpendiculars and Bisectors
Day 1 Part 1
CA Standard 16.0
Warmup
Simplify. 1. 6x + 11y – 4x + y 2. -5m + 3q + 4m – q 3. -3q – 4t – 5t – 2p 4. 9x – 22y + 18x – 3y 5. 5x2 + 2xy – 7x2 + xy
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB,
then CA = CB.
A P B
C
Converse of the Perpendicular Bisector Theorem
In the diagram shown, MN is the perpendicular bisector of ST.
What segment lengths in the diagram are equal?
Explain why Q is on MN
T
M N
S
Q
12
12
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If m<BAD = m<CAD, then DB = DC.
A
B
C
D
Converse of the Angle Bisector Theorem
Use the diagram to answer the following. In the diagram, F is on the bisector of
< DAE. If m<BAF = 50, then m<CAF = ____ If FC = 10, then FB = ____ Is triangle ABF congruent to triangle ACF?
Explain. A
B
D
G
F
C
E
5.2 Bisectors of a Triangle
Day 1 Part 2
CA Standards 16.0, 21.0
In the figure, YW bisects <XYZ.
m<XYZ = 6x + 2, m<ZYW = 8x – 6.
Solve for x and find m<XYZ.
Y
X
W
Z
Concurrency of Perpendicular Bisectors of a
Triangle The perpendicular bisectors of a triangle
intersect at a point
that is equidistant from
the vertices of the
triangle.
PA = PB = PCA
B
C
P
P is also called the circumcenter of the triangle.
Use the diagram shown.
E is the circumcenter of Δ ABC. DA = ___ BF = ___ <EFC = ___
A
D
B
E
CF
Definitions
Concurrent lines: three or more lines intersect in the same point.
Point of concurrency: the point of intersection of the lines.
Concurrency of Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
PD = PE = PF
A
B
C
P
E
D F
The point of concurrency can be inside the triangle, on the triangle, or outside of the triangle. Acute Triangle: inside Right Triangle: on Obtuse Triangle: outside
Example
Which segments
are congruent?
M
P N
RQ
L
S
Use the diagram shown.
E is the circumcenter of Δ ABC. DA = ___ BF = ___ <EFC = ___
A
D
B
E
CF
Mini quiz on definitions…
The _____________ of the angle bisectors is called the incenter of the triangle.
If three or more lines intersect at the same point, the lines are ________.
The point of concurrency of the perpendicular bisectors of a triangle is called ____________________.
Point of concurrency
Concurrent
Circumcenter of the triangle
Construction
Pg. 268 # 14, 15 Pg. 275 # 5 – 9
Pg. 269 # 21 – 29, 32 Pg. 275 # 10 - 22
5.3 Medians and Altitudes of a Triangle
Day 2 Part 1
CA Standards 16.0
Warmup
Find BD.
12 12
15
C
D
B A
AC = ___ m<DCB = ___ m<B = ___
A
B
C20
L
D
55
20
55
35
Median of a triangle.
Median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Median
The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle.
•P
Centroid
Concurrency of Medians of a Triangle
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.
AP = 2/3 AD
BP = 2/3 BF
CP = 2/3 CE
A B
C
DF
E
P
P is the centroid of ∆QRS shown. Find RT and RP when PT = 5.
R
SQ T
P
Sketch ∆JKL with J(7,10), K(5,2), L(3,6). Find the coordinates of the centroid of ∆JKL.
Altitude of a triangle
An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.
Altitude
Every triangle has three altitudes.
The lines containing the altitudes are concurrent and intersect at a point: orthocenter of the triangle.
Where is the orthocenter located in each type of triangle? Acute triangle Right triangle Obtuse triangle
Use the diagram shown and the given information to decide in each case whether EG is a
perpendicular bisector, an angle bisector, a median, or an altitude of Δ DEF.
1. DG = FG
2. EG DF
3. m<DEG = m<FEGT
E
D G F
Median
Altitude
Angle bisector
The angle bisectors of Δ ABC meet at point D. Find DE.
A
B
C
D
E F
G
LL19
28
19
5.4 Midsegment Theorem
Day 2 Part 2
CA Standards 17.0
Review
Given PQ = 14, SU = 6, and QU = 3, find the perimeter of Δ STU.
Q
U
R
S
T P
Midsegment Theorem
The segment connecting the midpoint of two sides of a triangle is parallel to the third side and is half as long.
DE ll AB and DE = ½ AB
A B
C
D E>
>
UW and VW are midsegment of Δ RST. Find UW and RT.
R
S
V
U
W
T812 6
16
GH, HJ and JG are midsegments of Δ EDF.
1. JH ll ___
2. EF = ___
3. DF = ___
4. ___ ll DE
5. GH = ___
6. JH = ___
D
G
F
H
EJ
24
810.6
DF
21.2
16
GH
12
8
Given the midpoints of a triangle are (7,4), (5,6) and (8,7), find the coordinates of the vertices.
Pg. 282 # 3 – 12 Pg. 283 # 17 – 20 Pg. 290 # 3 – 22, 26 – 29
5.5 Inequalities in One Triangle
Day 3 Part 1
CA Standards 6.0, 13.0
Warmup
Solve the inequality. 1. -x + 2 > 7 2. c – 18 < 10 3. -5 + m < 21 x – 5 > 4 z + 6 > -2
Theorems
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.
m<B > m<CA
B
C
3 7
List the angles in order from greatest to least.
A
B C
27
23
18
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 angle.
D
E
F
60°
40°
EF > DF
Write the measurements of the triangles in order from least to greatest.
J
H
G
100°
35°
45°
Q
R
P
5
6
7
JG, HJ, HG m<R, m<Q, m<P
List the sides in order from longest to shortest.
G E
F
45° 65°
Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent (not next to) interior angles.
A
B C1
m< 1 > m<A
m<1 > m<B
Constructing a Triangle
Construct a triangle with the given group of side length, if possible. 4 in, 4 in, 4 in
2 in, 4 in, 6 in
3 in, 4 in, 5 in
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
AB + BC > AC
AC + BC > AB
AB + AC > BCA
B
C
Use the diagram to solve the inequality
AB + BC > AC.
A
B
C
6x + 3
4x + 5
3x + 2
Two sides of a triangle have lengths 4 and 14. Describe the possible length of the third side.
Two sides of a triangle have lengths 10 and 15. Describe the possible length of the third side.
5.6 Indirect Proof and Inequalities in Two Triangles
Day 3 Part 2
CA Standards 2.0, 6.0, 13.0
Review
1. What is the sum of x and y?
2. Which measure is greater, x ° or y °?
x
y21
20
Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.
R
S
T
W
V
X100° 80°
RT > WX
Use Hinge theorem to complete the blank with <, >, or =.
1. RS __ TU 2. m<1 ___ m< 2
3. XY __ ZY
R
S
T
U110°
130°
1
2
15 13
< >
X
Y
Z
41°
38°
<
Converse of the Hinge Theorem
What is the largest angle that is part of a triangle?
9910
44
W
XY Z
Review List the sides in order from shortest to
longest.L
M
N
75°
74°
A
B
C50°
49°
75° + 74°+ m<N = 180°149° + m<N = 180°m<N = 31°
31°
LM, LN, MN
49 ° + 50 ° + m<B = 180 °99 ° + m<B = 180 °m<B = 81 °
81 °
BC, AB, AC
Review
Solve for a
2a + 7
a + 19
Extra Credit!!
Use the diagram to solve.
1. Find the value of x.
2. Find m<B
3. Find m<C
4. Find m<BAC
D
A
BC
3x°
(x+13)°(x+19)°
Pg. 298 # 6 – 25, 34 Pg. 305 # 3 – 23, 26, 27
Ch 5 Review
Day 4 Part 1
Warmup
Review
Find the value of x.
40
x
2x+6
Review
Ch 5 Culminating Task
Ch 5 Test
Day 4 Part 2