The document discusses topics in coordinate geometry including slopes, length of line segments, midpoints, and proofs. It provides formulas and examples for calculating slopes, lengths of lines, and midpoints. It also discusses using an analytical approach to prove geometric theorems through using coordinates, formulas, and algebraic manipulations rather than relying solely on diagrams.

1. Coordinate Geometry

2. Analytic Geometry
Coordinates
Slopes
Midpoints
Length of line segments

3. Slopes/Gradients
Slope = (change in y) / (change in x)
x
y



A(x2,y2)
B(x1,y 1)
12
12
xx
yy



y2 – y1
x2 – x1

4. Example
A(6, 4)
B(-7, -1)
12
12
xx
yy



)7(6
)1(4



13
5


5. Example
If the slope of the line joining A(-3,-2) and
B(4, y) is – 6, calculate y.
12
12
xx
yy
m



)3(4
)2(
6



y
7
2
6


y
-42 = y + 2  y = - 44

6. Length of line segment
AB 2 = BC 2 + AC 2
= ( x 2 – x 1 ) 2 + ( y 2 – y1 )2
A(x2, y2)
B(x1, y1)
y2 – y1
x2 – x1
C
AB =    2
12
2
12 yyxx 

7. Distance Formula
Subtract the first x from the second
do the same with y
Square them both and add together,
do not multiply
Take the square root of what you got
and plug it in
If you got the right answer, then you
win!

8. Length of line segments
Determine the length of the line joining the
points X( 6,4) and Y( -2,1)
   22
14)2(6 
AB =    2
12
2
12 yyxx 
   22
38 
73

9. Determine x if the length of line joining
A(x,1) and B( -1, 3) is 2 2
   22
31)1(22  x
AB =    2
12
2
12 yyxx 
   22
2122  x
4 = (x + 1)2
2 = x + 1
8 = (x + 1)2 + 4
x + 1 = 2
x = 1
x + 1 = - 2
x = -3

10. The Midpoint Formula
The midpoint is easy to find
Take both the x’s and combine
Do the same for the y’s and divide
each by two
There is the midpoint formula for
you.

11. Midpoint of line segments
2
12 xx 
A(x2, y2)
B(x1, y1)
( , )
2
12 xx 
2
12 yy 
C
2
12 yy 

12. Midpoint of line segments
Give the coordinates of the midpoint of
the line joining the points A(-2, 3) and
B(4, -3)
2
12 xx ( , )
2
33
2
12 yy 
2
42 
( , )
(1, 0)

13. Analytical Way of
Proving Theorems

14. The Role of Proof in
Mathematics

15. “For a non-believer,
no proof is sufficient…
For a believer, no
proof is necessary…”

16. Proof
Convincing demonstration that
a math statement is true
To explain
Informal and formal
Logic
No single correct answer

17. ANALYTIC PROOFS
Analytic proof – A proof of a geometric
theorem using algebraic formulas such
as midpoint, slope, or distance
Analytic proofs
pick a diagram with coordinates that
are appropriate.
decide on formulas needed to reach
conclusion.

18. Preparing to do analytic proofs

19. Preparing analytic proofs
 Drawing considerations:
1. Use variables as coordinates, not (2,3)
2. Drawing must satisfy conditions of the
proof
3. Make it as simple as possible without
losing generality (use zero values, x/y-
axis, etc.)
 Using the conclusion:
1. Verify everything in the conclusion
2. Use the right formula for the proof

23. Good to know!
Q.E.D. is an initialism of the Latin
phrase quod erat demonstrandum,
meaning "which had to be demonstrated".
The phrase is traditionally placed in its
abbreviated form at the end of a
mathematical proof or when what was
specified in the setting-out — has been
exactly restated as the conclusion of the
demonstration.

27. Prove that the diagonals of a
parallelogram bisect each
other.
 STEP 1: Recall the definition
of the necessary terms.
 STEP 2: Plot the points.
 Choose convenient
coordinates.

28. Prove that diagonals of a
parallelogram bisect each other.

29. (0, 0) (a, 0)
(b, c) (a +b, c)
To prove that the
diagonals of a
parallelogram bisect
each other,
their __________
must be shown to
be _________.

30. (0, 0) (a, 0)
(b, c)
(a +b, c)
O
B
C
A
Let E and F be the
midpoint of diagonals
𝑂𝐶 and 𝐵𝐴.
E = (
𝑎+𝑏
2
,
𝑐
2
)
F = (
𝑎+𝑏
2
,
𝑐
2
)
Therefore, the diagonals of a
parallelogram bisect each other.

31. Prove that a parallelogram whose
diagonals are perpendicular is a
rhombus.

32. Two lines are
perpendicular if
the product of
their slopes is -1.
Slope of diagonal
𝐵𝐴 is
𝑐
𝑏 −𝑎
.
Slope of diagonal
𝑂𝐶 is
𝑐
𝑎+𝑏
.
Rhombus is a
parallelogram with all
sides congruent.

33. Slope of diagonal
𝐵𝐴 is
𝑐
𝑏−𝑎
.
Slope of diagonal
𝑂𝐶 is
𝑐
𝑎+𝑏
.
𝑐
𝑏 − 𝑎
∙
𝑐
𝑎 + 𝑏
= −1
c2= -(b – a)(a + b)-(b2 – a2)
c2= a2 – b2

34. OA = a
OB = 𝑏2 + 𝑐2
BC = (𝑎 + 𝑏 − 𝑏)2
BC = 𝑎2𝑎
AC = (𝑎 + 𝑏 − 𝑎)2+𝑐2AC = 𝑏2 + 𝑐2
c2= a2 – b2
OB = 𝑏2 + 𝑎2 − 𝑏2OB = 𝑎2𝑎
AC = 𝑎2𝑎
Therefore, the parallelogram is a
rhombus.

35. Prove that in any triangle, the line
segment joining the midpoints of two
sides is parallel to, and half as long as
the third side.
(0, 0)
(a, 0)
(b, c)

36. CENTER – RADIUS FORM of the CIRCLE
    222
rkyhx 
The center of the circle is at (h, k).
    1613
22
 yx
The center of the circle is (3,1) and radius
is 4
Find the center and radius and graph this circle.
This is r 2 so r = 4
2-
7
-
6
-
5
-
4
-
3
-
2
-
1
1 5 73
0
4 6 8

37. Recall:
Square of a Binomial:
(x  a)² = x²  2ax + a²
Example: (x + 4)² + (y – 2)2= 25
(x + 4)2

38. Recall:
Square of a Binomial:
(x  a)² = x²  2ax + a²
Example: (x + 4)² + (y – 2)2= 25
(y - 2)2

39. 034622
 yxyx
We have to complete the square on both the x's and
y's to get in standard form.
______3____4____6 22
 yyxx
Group x terms and a
place to complete the
square
Group y terms and a
place to complete the
square
Move constant
to the other
side
9 94 4
    1623
22
 yx
Write in factored form, the standard form.
Find the center and radius of the circle:
So the center is at (-3, 2) and the radius is 4.