Download 4-8_Triangles_and_Coordinate_Proof

Five-Minute Check (over Lesson 4–7)
CCSS
Then/Now
New Vocabulary
Example 1: Position and Label a Triangle
Key Concept: Placing Triangles on Coordinate Plane
Example 2: Identify Missing Coordinates
Example 3: Write a Coordinate Proof
Example 4: Real-World Example: Classify Triangles
Over Lesson 4–6
Identify the type of congruence
transformation shown as a
reflection, translation, or
rotation.
A. reflection
B. translation
C. rotation
Over Lesson 4–6
Identify the type of congruence
transformation shown as a
reflection, translation, or
rotation.
A. reflection
B. translation
C. rotation
Over Lesson 4–6
Graph triangles LMN and PQR
with vertices L(–4, 5),
M(–4, 1), N(0, 3), and P(1, –1),
Q(1, –5), and R(5, –3). Then
identify the transformation.
A. reflection
B. translation
C. rotation
Over Lesson 4–6
Rectangle RSTU has vertices at (0, 0), (0, 4), (6, 4),
and (6, 0). Which of the following is a vertex of the
rectangle reflected over the x-axis?
A. (–6, 0)
B. (–4, –6)
C. (0, –4)
D. (–4, 0)
Content Standards
G.CO.10 Prove theorems about triangles.
G.GPE.4 Use coordinates to prove simple
geometric theorems algebraically.
Mathematical Practices
3 Construct viable arguments and critique
the reasoning of others.
2 Reason abstractly and quantitatively.
You used coordinate geometry to prove
triangle congruence.
• Position and label triangles for use in
coordinate proofs.
• Write coordinate proofs.
• coordinate proof
Position and Label a Triangle
Position and label right triangle XYZ with leg
d units long on the coordinate plane.
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-coordinate is d because
the base is d units long.
Position and Label a Triangle
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:
Which picture on the following slide would be the
best way to position and label equilateral triangle
ABC with side
w units long on the coordinate
plane?
A.
B.
C.
D.
Identify Missing Coordinates
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 ΔQRS 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 ΔABC.
A. A(d, 0); C(0, 0)
B. A(0, f); C(0, 0)
C. A(0, d); C(0, 0)
D. A(0, 0); C(0, d)
Write a Coordinate Proof
Write a coordinate proof to prove that the segment
that joins the vertex angle of an isosceles triangle to
the midpoint of its base is perpendicular to the base.
The first step is to position and label an isosceles triangle
on the coordinate plane. Place the base of the isosceles
triangle along the x-axis. Draw a line segment from the
vertex of the triangle to its base. Label the origin and label
the coordinates, using multiples of 2 since the Midpoint
Formula takes half the sum
of the coordinates.
Given: ΔXYZ is isosceles.
Prove:
Write a Coordinate Proof
Proof: By the Midpoint Formula, the coordinates of W,
the midpoint of
, is
The slope of
or undefined. The
slope of
therefore,
is
.
Finish the following 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
?
because ____.
A. their slopes are opposite.
B. the sum of their slopes is zero.
C. the product of their slopes is –1.
D. the difference of their slopes is 2.
is
or –1,
Classify Triangles
DRAFTING Write a coordinate proof to prove that the
outside of this drafter’s tool is shaped like a right
triangle. The length of one side is 10 inches and the
length of another side is 5.75 inches.
Classify Triangles
Proof: The slope of
or undefined. The slope of
or 0, therefore
ΔDEF is a right triangle.
The drafter’s tool is shaped like a
right triangle.
FLAGS Tracy wants to write a coordinate proof to
prove this flag is shaped like an isosceles triangle.
The altitude is 16 inches and the base is 10 inches.
What ordered pair should she use for point C?
A. (10, 10)
B. (10, 5)
C. (16, 10)
D. (16, 5)