Transcript Lesson 1 Contents - Headlee's Math Mansion

Lesson 4-7
Triangles and Coordinate Proof
5-Minute Check on Lesson 4-6
Transparency 4-7
Refer to the figure.
1. Name two congruent segments if 1  2.
2. Name two congruent angles if RS  RT.
3. Find mR if mRUV = 65.
4. Find mC if ABC is isosceles with AB  AC and mA = 70.
5. Find x if LMN is equilateral with LM = 2x – 4, MN = x + 6,
and LN = 3x – 14.
6.
Find the measures of the base angles of an
isosceles triangle if the measure of the vertex angle is 58.
Standardized Test Practice:
A
32
B
58
C
61
D
122
5-Minute Check on Lesson 4-6
Transparency 4-7
Refer to the figure.
1. Name two congruent segments if 1  2.
UW  VW
2. Name two congruent angles if RS  RT.
S  T
3. Find mR if mRUV = 65. 50
4. Find mC if ABC is isosceles with AB  AC and mA = 70. 55
5. Find x if LMN is equilateral with LM = 2x – 4, MN = x + 6,
and LN = 3x – 14.
10
6.
Find the measures of the base angles of an
isosceles triangle if the measure of the vertex angle is 58.
Standardized Test Practice:
A
32
B
58
C
61
D
122
Objectives
• Position and label triangles for use in
coordinate proofs
• Write coordinate proofs
Vocabulary
• Coordinate proof – uses figures in the
coordinate plane and algebra to prove
geometric concepts.
Classifying Triangles
…. Using the distance formula
y
D
Find the measures of the sides of ▲DEC.
Classify the triangle by its sides.
E
D (3, 9)
E (3, -5)
C
C (2, 2)
x
EC =
=
=
=
√ (-5 – 2)2 + (3 – 2)2
√(-7)2 + (1)2
√49 + 1
√50
DC =
=
=
=
√ (3 – 2)2 + (9 – 2)2
√(1)2 + (7)2
√1 + 49
√50
ED =
=
=
=
=
√ (-5 – 3)2 + (3 – 9)2
√(-8)2 + (-6)2
√64 + 36
√100
10
DC = EC, so ▲DEC is isosceles
Position and label right triangle XYZ with leg
long on the coordinate plane.
d units
Use the origin as vertex X of the triangle.
Place the base of the
triangle along the positive
x-axis.
Position the triangle in the
first quadrant.
Since Z is on the x-axis, its
y-coordinate is 0. Its
X (0, 0)
x-coordinate is d because
the base is d units long.
Z (d, 0)
Since triangle XYZ is a right triangle the
x-coordinate of Y is 0. We cannot determine the
y-coordinate so call it b.
Answer:
Y (0, b)
X (0, 0)
Z (d, 0)
Position and label equilateral triangle ABC with side
w units long on the coordinate plane.
Answer:
Name the missing coordinates of isosceles right
triangle QRS.
Q is on the origin, so its coordinates
are (0, 0).
The x-coordinate of S is the same
as the x-coordinate for R, (c, ?).
The y-coordinate for S is the distance
from R to S. Since QRS is an
isosceles right triangle,
The distance from Q to R is c units.
The distance from R to S must be
the same. So, the coordinates of S
are (c, c).
Answer: Q(0, 0); S(c, c)
Name the missing coordinates of isosceles right ABC.
Answer: C(0, 0); A(0, d)
Write a coordinate proof to prove that
the segment drawn from the right angle
to the midpoint of the hypotenuse of an
isosceles right triangle is perpendicular
to the hypotenuse.
Proof: The coordinates of the midpoint D are
The slope of
or 1. The slope of
therefore
.
is
or –1,
FLAGS Write a coordinate proof to prove this flag is
shaped like an isosceles triangle. The length is 16
inches and the height is 10 inches.
C
Proof: Vertex A is at the origin and B is at (0, 10). The
x-coordinate of C is 16. The y-coordinate is halfway
between 0 and 10 or 5. So, the coordinates of C are
(16, 5).
Determine the lengths of CA and CB.
Since each leg is the same length, ABC is isosceles. The
flag is shaped like an isosceles triangle.
Summary & Homework
• Summary:
– Coordinate proofs use algebra to prove
geometric concepts.
– The distance formula, slope formula, and
midpoint formula are often used in coordinate
proofs.
• Homework: Chapter Review handout