Coordinate Geometry

Analytic Geometry

CoordinatesSlopesMidpointsLength of line segments

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

Example

A(6, 4)

B(-7, -1)

12

12

xx

yy

)7(6

)1(4

13

5

 

Example

If the slope of the line joining A(-3,-2) and B(4, y) is – 6, calculate y.

12

12

xx

yym

)3(4

)2(6

y

7

26

y

-42 = y + 2 y = - 44

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 – x1C

AB = 2122

12 yyxx

Distance Formula

Subtract the first x from the second do the same with y Square them both and add together, do not multiplyTake the square root of what you got and plug it inIf you got the right answer, then you win!

Length of line segmentsDetermine the length of the line joining the

points X( 6,4) and Y( -2,1)

22 14)2(6

AB = 2122

12 yyxx

22 38

73

Determine x if the length of line joining

A(x,1) and B( -1, 3) is 2

22 31)1(22 x

AB = 2122

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

The Midpoint Formula

The midpoint is easy to findTake both the x’s and combineDo the same for the y’s and divide each by twoThere is the midpoint formula for you.

Midpoint of line segments

212 xx

A(x2, y2)

B(x1, y1)

( , )

212 xx

212 yy

C

212 yy

Midpoint of line segmentsGive the coordinates of the midpoint of the line joining the points A(-2, 3) and B(4, -3)

212 xx ( , )

2

332

12 yy

2

42 ( , )

(1, 0)

Analytical Way of Proving Theorems

The Role of Proof in Mathematics

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

Proof Convincing demonstration

that a math statement is true To explain Informal and formal Logic No single correct answer

ANALYTIC PROOFSAnalytic proof – A proof of a geometric

theorem using algebraic formulas such as midpoint, slope, or distance

Analytic proofspick a diagram with coordinates that

are appropriate.decide on formulas needed to reach

conclusion.

Preparing to do analytic proofs

Preparing analytic proofs Drawing considerations:

1. Use variables as coordinates, not (2,3)2. Drawing must satisfy conditions of the

proof3. Make it as simple as possible without

losing generality (use zero values, x/y-axis, etc.)

Using the conclusion: 1. Verify everything in the conclusion2. Use the right formula for the proof

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.

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.

Prove that diagonals of a parallelogram bisect each other.

(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 _________.

(0, 0) (a, 0)

(b, c)(a +b, c)

O

BC

A

Let E and F be the midpoint of diagonals and .

E = (, )

F = (, )

Therefore, the diagonals of a parallelogram bisect each other.

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

Two lines are perpendicular if the product of their slopes is -1.

Slope of diagonal is .

Slope of diagonal is .

Rhombus is a parallelogram with all sides congruent.

Slope of diagonal is .

Slope of diagonal is .

c2= -(b – a)(a + b) -(b2 – a2)

c2= a2 – b2

Rhombus is a

parallelogram with

all sides congruent.

OA = a

OB =

BC = BC =

AC = AC =

c2= a2 – b2

OB = OB = 𝑎

AC =

Therefore, the parallelogram is a rhombus.

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)

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

7

3

0

4

6

8

Recall:

Square of a Binomial:(x a)² = x² 2ax + a²Example: (x + 4)² + (y – 2)2= 25

(x + 4)2

Recall:

Square of a Binomial:(x a)² = x² 2ax + a²Example: (x + 4)² + (y – 2)2= 25

(y - 2)2

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 yxWrite 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.