Download Note Sheet 4-8

Geometry 4-8 Triangles and Coordinate Proof
A coordinate proof
uses figures drawn in the coordinate
plane and algebra to prove geometric concepts.
There are four basic rules for placing your figures in the
coordinate plane:
Use the origin
as a vertex or center of the triangle.
Place at least one side of a triangle on an axis .
Keep the triangle in the first quadrant
if possible.
Use coordinates
that make computations simple.
Name the coordinates of isosceles triangle XYZ.
Y(?,?)
X(?,?)
Z(a,?)
X is at the origin, so X is (0,0).
Z is on the x-axis, so Z is (a,0).
Y is halfway between X and Z, so
the x-coordinate is a/2. We don't
know the y-coordinate, so we'll
call it b. So Y is (a/2, b).
Write a coordinate proof to show that a line segment joining
the midpoints of two sides of a triangle is parallel to the
third side.
2b + 0 2c + 0
M=
= (b,c)
2 , 2
B(2b,2c)
2b + 2a 2c + 0
M
N=
= (b + a,c)
N
2 , 2
c-c
0
m(MN) = b + a - b = a = 0
A
C
(0,0)
(2a,0)
m(AC) = 0 - 0 = 0 = 0
2a - 0
2a
Since the slopes are equal, MN || AC.
(
(
)
)
Position and label isosceles triangle JKL on the coordinate
plane so that its base JL is a units long, vertex K is on the
y-axis, and the height of the triangle is b units.
K (0,b)
(-1/2 a,0) J
L(1/2 a,0)
PAUSE HERE AND TRY THIS ONE.
Name the coordinates of isosceles right triangle ABC.
A(?,?)
B(?,?)
C(a,?)
B is at the origin, so B is (0,0).
C is on the x-axis, so C is (a,0).
The triangle is isosceles, and A is
on the y-axis, so A is (0,a).