Analytic Proof
1.2 Analytic Proof
One way to prove some geometric theorems is to make use of, or refer to, a coordinate system. Such a proof is called an analytic proof .
d
Labeling Points
(a, 0)(0, 0)
(0, b)
c(d, c)
23a
,2a
2a
Labeling Points
(a, 0)(0, 0)
Equilateral triangle
23a
a
C( , )da
Give the Missing Coordinates
D(d , )
A(0, 0)
Rhombus
22 da
B(a, 0)
22 da 1 32
general triangle isosceles triangle
Triangles
right triangle
3
equilateral triangle
Triangles
trapezoid
isosceles trapezoidgeneral quadrilateral
Quadrilaterals
parallelogram rhombus
22 da 22 da
Quadrilaterals
squarerectangle
Quadrilaterals
Tools to use in analytic proving:
distance formula;
midpoint formula;
slope formula;
theorems on parallelism and perpendicularity; and
some algebraic means.
Main parts of an analytic proof:
diagram (completely labeled)
objective
proof proper (computations)
conclusion
A(0, 0)
Example 1: Prove analytically that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
Solution:
Consider theright triangle with vertices A, B and C and D be the midpoint of the hypotenuse.
B(0, b)
C(a, 0)
D(a/2, b/2)
.CDBDAD We need to SHOW that,
AD
22
2b
2a
4b
4a 22
22
02b
02a
Solution (continuation)…
A(0, 0)
B(0, b)
C(a, 0)
D(a/2, b/2)
CD
2 2a b
2 2
4b
4a 22
22
2b
02a
a
Solution (continuation)…
A(0, 0)
B(0, b)
C(a, 0)
D(a/2, b/2)
BD
4a
4b 22
22
2a
02b
b
Solution (continuation)…
A(0, 0)
B(0, b)
C(a, 0)
D(a/2, b/2)Hence, the midpoint D of the hypotenuse is equidistant from the vertices A,B, and C.
A(0, 0) B(a, 0)
Example 2: Prove analytically that if the diagonals of a parallelogram are congruent then the parallelogram is a segment or a rectangle.
Solution:Consider the parallelogram with vertices A, B, C, and D and with diagonals AC .BDand
D(d, c) C(a+d, c)
Solution (continuation)…
It was assumed that . But,AC BD .
AC a d c2 2
A(0, 0) B(a, 0)
D(d, c) C(a+d, c)
BD a d c2 2 . and
a d c a d c2 22 2
a d a d2 2
a ad d a ad d2 2 2 22 2
ad4 0
ad 0a or d0 0
Solution (continuation)…
If ,0a then point B has coordinates (0,0) also.
If ,0d then the vertices of the parallelogram are (0,0), (a,0), (a,c) and (0,c).
Hence, if the diagonals of a parallelogram are congruent then the parallelogram is a segment or a rectangle.
Example 3: The points A(3,-2), B(4,1) and C(-3,5) are vertices of a triangle. Show that the line through the midpoints D and E of the sides AB and AC , respectively, is parallel to the third side BC of the triangle. Also show that
BC2
1DE
DO NOT FORGET TO…
look how other geometric figures were labeled;
prove more geometric theorems using the techniques presented;
study other examples.
End of 1.2
Quiz 1 Close your notes and handouts.
Bring out 2 quarter sheets of paper.
Write your name and recitation section correctly.
This quiz is good for 10 minutes only and worth 10 points.