Rene Descartes (left) and Pierre de Fermat (right) developed analytical geometry independently of each other in the 1600 s.

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Lesson 14 Analtical Geometr, Part I Rene Descartes (left) and Pierre de Fermat (right) developed analtical geometr independentl of each other in the 1600 s.

- ordered pair: A pair of numbers, written in a specific order. On a coordinate plane, an ordered pair is used to identif the location of a point, and has the form (,).

1 Lesson 14 Analtical Geometr, Part I Rene Descartes (left) and Pierre de Fermat (right) developed analtical geometr independentl of each other in the 1600 s. Rules and Definitions Rules No new rules for Lesson 14. Definitions - analtical geometr: Where algebra and geometr meet on the coordinate plane. - ordered pair: A pair of numbers, written in a specific order. On a coordinate plane, an ordered pair is used to identif the location of a point, and has the form (,). - cartesian coordinate sstem: Also called a coordinate plane, it is a plane containing a horizontal, -ais and vertical, -ais. It is used to graph ordered pairs, functions, eperimental data, etc. 14A Foundations of Analtical Geometr Rene Descartes ( ) and Pierre de Fermat ( ) developed analtical geometr independentl of each other in the earl 1600 s. At this time, people were using maps to eplore the Earth with greater frequenc. Locations on maps were identified using east-west lines of latitude and north-south lines of longitude. Analtical geometr takes this idea and uses it to identif the location of a mathematical point (Lesson 9) on a cartesian coordinate sstem. Although Fermat actuall described analtical geometr first, Descartes published first, so he is usuall credited with discovering analtical geometr. Descartes is also famous for the phrase Cogito, ergo sum (I think, therefore I am). While Descartes firml believed in God s eistence, his philosoph was that

2 we could reason apart from God. Certainl, we can think about things without having to insert the word God in ever sentence! That s not the problem. The problem is when we think about things without ever acknowledging God or epressing thankfulness for the abilit He gave us to reason in the first place (Romans 1:20-21). The Christian sas Th Word is truth (John 17:17). Descartes said his own eistence and thoughts are more fundamental truths. God reveals knowledge to us through His word (special revelation) and His works (general revelation). We ma think we are concluding God eists based on our eperiences, but in realit, God reveals Himself to us first, then we conclude He eists. It is because of Him, not because of us, that we can know God and have a relationship with Him. This is where Descartes went wrong. It is not He loves us because we first loved Him, it s We love Him because He first loved us (I John 4:19). As Christians, we must tr hard to understand His attribute of unit and diversit, perfectl represented in the Godhead, where God = Father, Son, Spirit (see Lessons 1A, 4A). The attribute eists everwhere, and reminds us to be cautious about how we go about separating or mingling things. For eample, the Pthagoreans erred when the said all is number, foolishl worshiping numbers as if the were gods. Men like Descartes, though, put too much effort into separating man and religion, falsel concluding that humans could reason about things apart from God. Others took Descartes ideas and removed God from the picture altogether. The false idea of separating reason from the Author of reason has led to much destruction. On a national level, the banning of Christianit b French revolutionaries in 1789 and the Soviets in 1917 eventuall led to mass paranoia and bloodshed in those nations. On an individual level, it has resulted in the murder of tens of millions of babies through abortion. Histor reveals that humans reason poorl when the reject God and His word. 14B The Coordinate Plane Building a coordinate plane: Much of analtical geometr takes place on the coordinate plane. To make a coordinate plane, draw two number lines, one perpendicular to the other. Locate 0 for both at their intersection. We will call the horizontal number line the -ais, and the vertical number line the -ais. The point of intersection is the origin.

3 1 Practice making some coordinate planes like the one shown. Use a straightedge, or draw them freehand. The hardest part is evenl spacing the tick-marks, but ou will improve with practice! Coordinate planes, or graphs often are drawn as a grid, making locating points easier. This is the form we will use most often in this course.

4 Eample 14.1 Graph and label the following ordered pairs: A=(2,3), B=(-1,3), C=(-3,-4) solution: To instruct someone to locate a point on a graph, the phrases graph a point or plot a point are often used. To graph an point, start at the origin and move in the -direction the given amount, then up or down in the -direction for the given amount, and draw a point. That s it! The path taken (arrows) for plotting point A is shown: Graphing points is reall eas. Graphing lines and curves is a little more difficult, and we will cover that in Lesson 16 and 17. Practice Set 14 (subscripts tell ou which lesson each problem came from) You do not need a calculator for an problems and Descartes developed analtical geometr independentl of each other in the 1600 s. A) Fermat B) Kepler C) Euler D) Pascal Unlike, the goal of Christians should not be to separate the Creator from His creation, but to stud His creation and acknowledge Him. A) Fermat B) Kepler C) Descartes D) Pascal Make our own coordinate plane. Then, graph the following ordered pairs and label them: A) (2,-4) B) (-1,-3) C) (-3,5)

5 4 14. Write the coordinates of the points shown on the graph below Add to our one-point perspective highwa drawing from 13B and problem Tr to add another rectangular building on the left side of the road, behind the building drawn in problem Circle the net ou think would fold to form the right solid shown below Is this an eample of a tesselation or a fractal. Wh?

6 8 12,11. Find and a. For, do not use a calculator. Just leave the answer in square root form Find and, then find c. Lines m, n, and t are parallel c 3 m n t 10 12, 11, 9. Use the properties of quadrilaterals to determine whether each statement is true. Is its contrapositive also true? a) If a quadrilateral has equal diagonals, then it is also a rectangle. b) If a quadrilateral is a rhombus, then it is also a parallelogram Given the central angle ABC = 136, find arcs AC and ADC Write a sllogism for the Euler diagram shown. 2-wheeled objects All biccles That vehicle

7 Use a compass and straightedge (or Geometer s Sketchpad) to construct an equilateral triangle. Then, rearrange the steps shown in the correct order to complete the proof of Euclid s Proposition 1. Don t rewrite everthing, just write the letters A,B,C, etc. in the correct order. A)With center A and distance AB, construct a circle BCD[Post. 3]. B) With center B and distance BA, construct a circle ACE[Post. 3]. C) Therefore, the three line segments CA, AB, and BC are equal to each other, and triangle ABC is therefore an equilateral triangle, being what was required to do. D) But CA was also proved equal to AB. Therefore, CA must also equal CB, since things equal to the same thing are equal to each other [Aiom 1]. E) Now, since point A is the center of circle CBD, AC = AB [Def. of circle]. F) Also, since point B is the center of circle ACE, BC = BA [Def. of circle]. G) From point C, where the circles intersect, construct line segments CA and CB[Post. 1]. H) Construct line segment AB[Given] Find and Which of the following is an obtuse angle? A) B) C) Factor. Find the GCF first Evaluate a 2 + 2b 2 if a = -1 and b = The dress cost $100, but was on sale for 20% off. What was the sale price? (Hint: think 80% of $100 equals what?) Simplif. 6i (-2) 2 (4-1) The crew gathered 8,742 peaches. If each bo held 12 peaches, how man boes were filled? How man peaches were left over?

LESSON 3.1 INTRODUCTION TO GRAPHING

LESSON 3.1 INTRODUCTION TO GRAPHING LESSON 3.1 INTRODUCTION TO GRAPHING 137 OVERVIEW Here s what ou ll learn in this lesson: Plotting Points a. The -plane b. The -ais and -ais c. The origin d. Ordered