Coordinate Geometry

Geometry - Coordinate Geometry:

check out this website:
http://regentsprep.org/Regents/mathb/1D/Coordinatelesson.htm

B(x₁, y₁) and A (x₂, y₂) is from the
Pythagorean Theorem which is:

c² = a² + b²

since a = (x₂ – x₁) and b = (y₂ – y₁)

we get

c² = (x₂ – x₁)² + (y₂ – y₁)²
so taking the square root of both sides we have our distance formula.

Here are the Slope Formula and the Midpoint Formula:

Using the following chart, if we have to prove congruent segments, we have to show they have equal length by using the distance formula.

Here is a web-site to hopefully help:

http://library.thinkquest.org/20991/geo/coordgeo.html

Geometry Proofs - unit 2 Triangles

Triangles – Proofs – Unit 2
1. Δ ABC is congruent to Δ ABC , this is by the Reflexive Postulate
2. if Δ ABC is congruent to Δ DEF then Δ DEF is congruent to Δ ABC, this is by the Symmetric Postulate
3. if Δ ABC is congruent to Δ DEF and Δ DEF is congruent to Δ GHI, Then Δ ABC is congruent to Δ GHI by the Transitivity Postulate

5 ways to show Triangle Congruence:
1. SAS = SAS Congruence Postulate - if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if AB = DE, BC = EF and angle B = angle E, then ΔABC = ΔDEF.

2. SSS = SSS Congruence Postulate - if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if AB = DE, BC = EF and AC = DF, then ΔABC = ΔDEF.

3. ASA = ASA Congruence Postulate - if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
Example: Given ΔABC and ΔDEF,
if angle A = angle D, AB = DE, and angle B = angle E, then Δ ABC = Δ DEF.

4. AAS = AAS Congruence Postulate - if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
Example: Given Δ ABC and Δ DEF,
if angle A = angle D, angle C = angle F, and BC = EF , then Δ ABC = Δ DEF.

5. HL = HL Congruence Postulate - if the leg and hypotenuse of one right triangle is congruent to the corresponding leg and hypotenuse of another right triangle, then the two triangles are congruent by hypotenuse - leg postulate.
Example: Given right Δ ABC and right Δ DEF, if angle B and angle E are both right angles and leg AB = leg DE and hypotenuse AC = hypotenuse DF, then Δ ABC = Δ DEF.

I) Vocabulary:
A) When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.
Example: Given Δ ABC is congruent to Δ PQR then we know
1) angle A = angle P, angle B = angle Q, and angle C = angle R
2) AB = PQ, BC = QR, and AC = PR By Corresponding Parts of Congruent Triangles are Congruent (CPCTC) - which means that if 2 triangles are congruent, then their corresponding parts are congruent

Make sure that you list the corresponding angles in the same order with the triangle congruence.
Example: ΔABC = ΔDEF is not the same as ΔABC = ΔEFD because
ΔABC = ΔDEF has angle A = angle D, angle B = angle E and angle C = angle F
while ΔABC = ΔEFD has angle A = angle E, angle B = angle F and angle C = angle D

B) Third Angles Theorem - if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

C) Reflexive Postulate - Every triangle is congruent to itself

D) Symmetric Postulate - If ΔABC = ΔDEF, then ΔDEF = ΔABC

E) Transitive Postulate - If ΔABC = ΔDEF and ΔDEF = ΔJKL, then ΔABC = ΔJKL

Theorems to remember:

If 2 angles of one triangle are congruent then the sides opposite are congruent.
If 2 sides of one triangle are congruent then the angles opposite are congruent.
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Practice Proofs:

Precalculus Unit 8 - chapter 6.3, 6.4, 10.5, 10.6, 10.7

PRECALCULUS
Unit 8
Vectors and Parametric/Polar Equations

Section 6.3A HW# 53; Pg.453; #1 – 45 by 4’s (1, 5, 9, …)

Section 6.3B; HW # 54; *Pg.453; #3, 11, 19, 23, 31, 39, 43
Pg.454; #49, 53, 57, 61, 65, 69, 73, 74, 82

Section 6.4; HW # 55; *Pg. 454; #51, 55, 59, 63, 67, 71,
Pg. 464; #1, 5, 9, 13, 17, 21, 25, 29, 33, 37
Quiz on Sections 6.3A – 6.3B Next Class

Section 10.5; HW# 56; *Pg.464; #3, 7, 11, 15, 19, 23, 27, 31
Pg.736; #1, 5, 9, 13, 21, 29, 41, 47

Section 10.6; HW# 57; *Pg.736; #3, 7, 11, 17, 27, 43
Pg.743; #5 – 41 by 4’s, 45, 50, 53 – 57 odds
Quiz on Sections 6.4 – 10.5 Next Class

Section 10.7; HW# 58; *Pg.743; #7, 11, 15, 23, 27, 39, 47, 59
Pg.752; #21 – 33 odds, 55, 59

Review HW #59
Pg.480; 39, 43, 47, 51, 59, 69, 79, 83, 85
Pg.762; 47, 49, 59, 65, 69, 73, 75, 81, 85, 89

HW #60; Unit 8 Test

Unit One for Proofs - Angles and Lines

The addition Postulate:

If a = b and c = d, then a + c = b + d

The Partition Postulate:

AB + BC = AC

These are two different concepts. Let's try a proof:

As you can see, in step 2, we added equal quantities to each other. This is Addition Postulate.

But in step 3, we added a part plus a part equals a whole so this is Partition.

Let's try another example:

Here we used the Subtraction Postulate: subtracting equal quantities from equal quantities.

In step 3, we had the whole minus a part equals a part. This is still Partition Postulate.

Geometry Vocabulary for Proofs

Vocabulary - Euclidean Geometry:

1. Space - is a set of points that forms a completely flat surface extending indefinitely in all directions.

2. Collinear Points - is a set of points all of which lie on the same straight line.

3. Coplanar points - is a set of points all of which lie on the same plane.

4. Betweenness of Points or Segment/Angle Addition Postulate (we will call this Partition Postulate) - point B is between point A and point C, if A, B and C are distinct collinear points and AB + BC = AC. We shall say that the part plus the part equals the whole (part + part = whole)

5. Segment or line segment - is a set of points consisting of two points on a line, called endpoints, and all the points on the line between the endpoints.

6. Length of a line segment - is the distance between the endpoints.

7. Congruent segments - are segments that have the same measure.

Congruent angles - are angles that have the same measure.

8. Midpoint of a segment - is the point of the line segment that divides the segment into two congruent segments or divides the line segment in half.

9. Bisector of a Segment or angle - is any line, ray, or point that intersects the segment or angle at its midpoint.

10. Rays - is a part of a line that consists of a point on the line called the endpoint and all the points on one side of the endpoint

11. Opposite Rays - are two rays of the same line with a common endpoint and no other point in common.

12. Angle - is a set of points that is the union of two rays having the same endpoint or vertex.

13. Sides of an angle - are the rays that make up the angle

14. Vertex of an angle - is the common endpoint of the two sides of an angle.

15. Adjacent angles - are two angle in the same plane that have a common vertex and a common side but do not have any common interior points.

16. Exterior sides of adjacent angles - the two sides of adjacent angles that are not common to both angles.

17. Vertical angles - are two angles in which the sides of one angle are opposite rays to the sides of the second angle. Vertical angles are congruent.

18. Addition Postulate - if A = B, then A + C = B + C

19. Subtraction Postulate - if A = B, then A - C = B - C

20. Multiplication Postulate - if A = B, then AC = BC

21. Division Postulate - if A = B and C does not equal zero, then A/C = B/C

22. Halves Postulate - if A = B, then A/2 = B/2

23. Doubles Postulate - if A = B, then 2A = 2B

24. Substitution Postulate - if A = 5 and A + B = 12, then 5 + B = 12.

- you plug in the equal value

25. Reflexive Postulate - A = A

26. Symmetric Postulate - if A = B, then B = A.

27. Transitivity Postulate - if A = B AND B = C, then A = C

OR if A is less than B AND B is less than C, then A is less than C.

28. Distributive Postulate - if A(B + C) then AB + AC

Other Vocabulary:

Congruent Angles - angles that have the same measure.

Complementary Angles - are two angles that the sum of their measures is 90 degrees

Supplementary Angles - are two angles that the sum of their measures is 180 degrees.

Perpendicular Lines - form right angles.

Right angle - has a measure of 90 degrees.

Linear pair - are two adjacent angles where their noncommon side are opposite rays.

Here is one way how to write a proof:

1. You set up a t-table and write the word Statements in the left column and Reasons in the right column.

2. You are given a statement with another statement to prove based on the given statement.

3. You write the given statement in the left column with the word given for your reason. Then you write statements that support what you are given with logically reason using vocabulary, postulates or already proven theorems.

Here are a few examples of algebraic proofs:

As you can see, you have been doing proofs, just not formally for awhile. We will continue these notes as we go along in our learning of proof writing. This is for a basic class in High School. There are many more ways to write proofs along with many more vocabulary words.

Precalculus Midterm Requirements

Requirements for Midterm:

Do 3 of the 6 projects from Chapters: Prerequisite, One, Two, Three, Four, or Five
These can be found at the end of each of the chapters.
Handwritten is fine. If you type the project out, one extra credit point per problem.
Show all work!! No work, no credit. Each question is worth 15 points.

Total points for midterm: 45 points.

The projects can be found on the following pages:
Prerequisite, page 70
Chapter 1, page 133
Chapter 2, page 213
Chapter 3, page 279
Chapter 4, page 373
Chapter 5, page 425

Answer all of the questions for each project chosen.

Precalculus Homework Unit 6 - chapter 5

Precalculus Homework Unit 6 - Chapter 5

5.1 Homework #38; Pg.381; 5, 7, 25, 27, 37, 43, 55, 59,
67, 71, 75-77odd, 91, 101, 103

5.2 Homework #39; Pg.389; 5-9odd, 21-25odd, 34, 35, 43, 55, 71;
*Pg. 381; 39, 41, 57, 69, 73, 93

5.3a Homework #40; Pg.400; 3, 11, 15, 23, 31, 39, 43, 47, 51, 55
*Pg. 389; 3, 19, 27, 37, 49, 59
Quiz on Sections 5.1-5.2 Next Class

5.3b Homework #41; Pg.400; 5, 7, 19, 27, 35, 41, 45, 53, 57, 59, 75, 76

5.4 Homework #42; Pg.408; 3, 5, 11, 15, 19, 25-27odd, 37-41odd, 45, 47 – 55 odd
*Pg 400; 7, 19, 27, 35, 43

5.5a Homework #43; Pg.418; 5, 7, 17, 19, 21-23odd, 31, 39,
43, 47, 51-57odd, *Pg 408; 21, 23, 35, 43, 57
Quiz on Sections 5.3-5.4 Next Class

5.5b Homework #44; Pg.418; 9, 18, 22, 27, 35, 45, 65, 71,
83-87odd, 103

Review Homework #45; Pg.422; 1 – 107 every 4th odd (1, 5, 9, …)

Homework #46; Chapter 5 Test