Lesson 4-7 Triangles and Coordinate
Proof
• Coordinate proof- uses figures in a coordinate
plane and Algebra to prove geometric concepts.
• Placing Figures in the Coordinate Plane
1. Use the origin as the vertex or the center of the
figure
2. Place at least one side of a polygon on an axis
3. Keep the figure within the 1st quadrant, if possible
4. Use coordinates that make math easy
Position and label right triangle XYZ with leg
plane.
d units long on the coordinate
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 ycoordinate is 0. Its
x-coordinate is d because the base is
d units long.
X (0, 0)
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 xcoordinate 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 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 a right 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: XYZ is isosceles.
Prove:
Proof: By the Midpoint Formula, the coordinates of W,
the midpoint of
, is
The slope of
or undefined. The
slope of
therefore,
is
.
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,
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.
Proof: The slope of
or undefined. The slope of
or 0, therefore
DEF is a right triangle.
The drafter’s tool is shaped like a
right triangle.
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.