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.