Page 101 - Math Course 3 (Book 2)

Coordinate Proofs

Find the Missing Coordinates Given: ΔXYZ is isosceles.
XW ≅ WZ
Example Prove: YW ⏊ XZ

Name the missing coordinates of isosceles right
triangle QRS. y Y(a, b)

y S(?, ?)

x
X(0, 0) W Z(2a, 0)

Q(?, ?) R(c, 0) x
Proof: By the Midpoint Formula, the coordinates
0 + 0
0 + 2a
of W, the midpoint of XZ, is ( , )
Q is on the origin, so its coordinates are (0, 0). 2 2
or (a, 0).
The x-coordinate of S is the same as the
0 – b
x-coordinate for R, (c, ?). The slope of YW is ( ) or undefined.
a – a
0 – 0
The slope of XZ is ( )or 0, therefore,
0 – 2a
The y-coordinate for S is the distance from R to S. YW ⏊ XZ.
Since ΔQRS is an isosceles right triangle, QR ≅ SR.
The distance from Q to R is c units. The distance
from R to S must be the same. So, the coordinates Classify Triangles
of S are (c, c).
Example

Answer Q(0, 0); S(c, c) 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
Coordinate Proof of another side is 5.75 inches.
y
Example E

(0, 10)

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. D F x
(5.75, 0)
The first step is to position and label an isosceles Proof: The slope of ED is ( )or
10 – 0
triangle on the coordinate plane. Place the base of 0 – 0
0 – 0
the isosceles triangle along the x-axis. Draw a line undefined. The slope of DF is( )
0 – 5.75
segment from the vertex of the triangle to its base. or 0, therefore ED ⏊ DF. ΔDEF is a right
Label the origin and label the coordinates, using
multiples of 2 since the Midpoint Formula takes triangle. The drafter’s tool is shaped like
half the sum of the coordinates.
a right triangle.

93