Post on 29-Apr-2018
transcript
Geometry Chapter 5 - Properties and Attributes of Triangles
Segments in Triangles
Lesson 1: Perpendicular and Angle Bisectors
● equidistant
Triangle congruence theorems can be used to prove theorems about equidistant points.
Distance and Perpendicular Bisectors
Theorem Hypothesis Conclusion
Perpendicular Bisector Theorem
If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of the
segment.
Converse of the Perpendicular
Bisector Theorem If a point is
equidistant from the endpoints of a
segment, then it is on the
perpendicular bisector of a segment.
● Locus
The perpendicular bisector of a segment can be defined as the locus of points in a plane that are
equidistant from the endpoints of the segment.
Applying the Perpendicular Bisector Theorem and Its Converse
Ex1: Find each measure
A. NM = B. BC = C. TU =
Remember that the distance between a point and a line is the length of the perpendicular segment from
the point to the line.
A BY
X
l
A B
X
l
Y
M
N
2.6 A
C
B
D
38
3812
3x + 9 7x - 17
U
T
Distance and Angle Bisectors
Theorem Hypothesis Conclusion
Angle Bisector Theorem
If a point is on the bisector of an
angle, then it is equidistant from the
sides of the angle.
Converse of the Angle Bisector
Theorem If a point in the interior of
an angle is equidistant from the sides
of the angle, then it is on the bisector
of the angle.
Applying the Angle Bisector Theorems
Ex2: Find each measure
A. BC = B. m∠EFH, given that
m∠EFG = 50°
C. m∠MKL
M
B
B
C
C
A
A
P
P
B
B
C
C
A
A
P
Ex4:
Write an equation in point-slope form for
the perpendicular bisector of the segment with
endpoints C(6, -5), and D(10, 1).
Geometry
Lesson 2: Bisectors of Triangles
Since a triangle has three sides, it has three perpendicular bisectors. When you construct the
perpendicular bisectors, you find that they have an interesting property.
● concurrent
● point of concurrency
● circumcenter of the triangle
Circumcenter Theorem
The circumcenter of a triangle is equidistant
from the vertices of the triangle.
The circumcenter can be inside the triangle, outside the triangle, or on the triangle.
Acute triangle Obtuse triangle Right triangle
P
P
C
C
B
B
A
The circumcenter of ΔABC is the center of its circumscribed circle.
● Circumscribed circle
•
Using Properties of Perpendicular Bisectors
Ex1: DG , EG , and FG are the
perpendicular bisectors of ΔABC. Find GC.
Ex2: Find the circumcenter of ΔHJK with
vertices H(0, 0), J(10, 0), and K(0, 6).
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also
concurrent.
● incenter of a triangle
Incenter Theorem
The incenter of a triangle is equidistant from the sides
of the triangle.
A
B
C
P•
P
P
C
C
B
B
A
Unlike the circumcenter, the incenter is always inside the triangle.
Acute triangle Obtuse triangle Right triangle
The incenter is the center of the triangle's inscribed circle.
● Inscribed circle
Using Properties of Angle Bisectors
Ex3: MP and LP are angle bisectors of
ΔLMN. Find each measure.
A. the distance from P to MN
.
B. m∠PMN
Ex4: A city planner wants to build a new library
between a school, a post office, and a hospital.
Draw a sketch to show where the library should be
placed so it is the same distance from all three
buildings.
Geometry
Lesson 3: Medians and Altitudes of Triangles
● median of a triangle
Every triangle has three medians, and the medians are concurrent.
● centroid of the triangle
The centroid is always inside the triangle. The centroid is also called the center of gravity because it is
the point where a triangluar region will balance.
A
B
C
P•
S L
P
A B
C
D
Centroid Theorem
The centroid of a triangle is located 2
3 of the distance from each vertex to the midpoint of the
opposite side.
*Remember, the centroid is closer to each side than to the verte
Using the Centroid to Find Segment Lengths
Ex1: In ΔLMN, RL = 21, and SQ = 4. Find
A. LS =
B. NQ =
Ex2: A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what
coordinates will the triangular region balance?
c
P
P
C
C
B
B
A
● altitude of a triangle
Every triangle has three altitudes. An altitude
can be inside, outside, or on the triangle.
● orthocenter of a triangle
Geometry
Lesson 4: The Triangle Midsegment Theorem
● midsegment of a triangle
midsegments:
midsegment triangle:
Every triangle has three midsegments, which form the midsegment triangle.
Q
RP
Examining Midsegments in the Coordinate Plane
Ex1: The vertices of ΔXYZ are X(-1, 8), Y(9, 2), and Z(3, -4). M and N are the midpoints of XZ YZ . Show that
A. MN // XY
B. MN = 1
2 XY.
The relationship shown in Example 1 is true for the midsegment of every triangle.
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to a side of the
triangle, and its length is half the length of that side.
A
C
B
Using the Triangle Midsegment Theorem
Ex2: Find each measure.
A. BD
B. m∠CBD
Ex3: In an A-frame support, the distance PQ is 46
inches. What is the length of the support ST if
S and T are at the midpoints of the sides?
Geometry
Relationships in TrianglesLesson 5: Indirect Proof and Inequalities in One Triangle
You have written proofs using direct reasoning. That is, you began with a true hypothesis and built a
logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that
the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof
is also called a proof by contradiction.
Writing an Indirect Proof
1. Identify the conjecture to be proven.
2. Assume the opposite (the negation) of the conclusion is true.
3. Use direct reasoning to show that the assumption leads to a contradiction.
4. Conclude that since the assumption is false, the original conjecture must be true.
Writing an Indirect Proof
Ex1A: Write an indirect proof that a right triangle cannot have an obtuse angle.
Ex1B:Write an indirect proof that if a > 0, then 1
a > 0.
Angle-Side Relationships in Triangles
Theorem Hypothesis Conclusion
If two sides of a triangle are not
congruent, then the larger angle
is opposite the longer side
If two angles of a triangle are not
congruent, then the longer side is
opposite the larger angle.
Ordering Triangle Side Lengths and Angle Measures
Ex2A: Write the angles in order from smallest to
largest.
B. Write the sides in order from shortest to
longest.
A triangle is formed by three segments, but not every set of three segments can form a triangle.
Triangle Inequality
The sum of any two side lengths of a triangle
is greater than the third length.
A B
C
X
Y
Z
A
B C
Applying the Triangle Inequality Theorem
Tell whether a triangle can have sides with the given lengths. Explain.
Ex1A: 3, 5, 7 B. 4, 6.5, 11 C. n + 5, n2 , 2n, when n = 3
Finding Side Lengths
Ex4: The lengths of two sides of a triangle are 8 in. and 13 in. Find the range of possible lengths for
the third side.
Ex5: The figure shows the approximate distances between cities in California. What is the range of
distances from San Francisco to Oakland?
Geometry
Lesson 6: Inequalities in Two Triangles
Inequalities in Two Triangles
Theorem Hypothesis Conclusion
Hinge Theorem If two sides of
one triangle are congruent to two
sides of another triangle and the
included angles are not
congruent, then the longer third
side is across from the larger
included angle.
A
D
B
C
E
F
Using the Hinge Theorem
Ex1A:Compare m∠BAC and
m∠DAC.
B: Compare EF and FG. C: Find the range of values for
k.
Ex2: John and Luke leave school at the same time. John rides his bike 3 blocks west and 4 blocks
north. Luke rides 4 blocks east and then 3 blocks at a bearing of N 10° E. Who is farther from
school?
Proving Triangle Relationships
Ex3: Write a two-column proof.
Given: AB ≅ CD ,
m∠ABD > m∠CDB
Prove: AD > CB
Statement Reason