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Geometry Applications: Points and Lines 1 TRANSCRIPT—Geometry Applications: Points and Lines 1 Geometry Applications: Points and Lines Geometry Applications: Points and Lines, Segment 1: Introduction OUR STUDY OF GEOMETRY OWES MUCH TO THE ANCIENT GREEKS. AND THERE IS NO BETTER SYMBOL OF ANCIENT GREECE THAN THE PARTHENON, A TEMPLE OVERLOOKING THE CITY OF ATHENS COMPLETED IN THE FOURTH CENTURY BC. YET A MORE PERMANENT MONUMENT WAS BUILT A CENTURY LATER BY THE MOST INFLUENTIAL MATHEMATICIAN OF ANCIENT GREECE, EUCLID. HIS BOOK, THE ELEMENTS, LAID THE GROUNDWORK FOR GEOMETRIC THINKING AND MATHEMATICAL REASONING. YET BOTH THE PARTHENON AND THE ELEMENTS SHARE A LOT IN COMMON WHEN IT COMES TO GEOMETRY. THE PARTHENON IS AN ARCHITECTURAL STRUCTURE AND ARCHITECTURE RELIES HEAVILY ON GEOMETRY. THE NOTIONS OF POINTS, LINES, PLANES, AND ANGLES ARE A KEY PART OF GEOMETRY AND OF THE PARTHENON. FOR EXAMPLE, TO SKETCH THE FRONT OF THE PARTHENON YOU WOULD DRAW A SERIES OF POINTS TRANSCRIPT—Geometry Applications: Points and Lines 2 TO MARK THE LINES FOR THE COLUMNS. YOU WOULD CONNECT THESE POINTS TO CONSTRUCT A TRIANGLE. FOR A THREE-DIMENSIONAL VIEW OF THE PARTHENON YOU WOULD NEED TO DRAW SEVERAL PLANES INDICATING THE VARIOUS LEVELS. GEOMETRIC CONSTRUCTIONS LIKE THESE RELY ON SOME UNDERLYING PRINCIPLES. IN THIS PROGRAM WE WILL COVER REAL-WORLD APPLICATIONS THAT EXPLORE THE FOLLOWING GEOMETRIC CONCEPTS: Geometry Applications: Points and Lines, Segment 2: Points IN THE SWISS COUNTRYSIDE SOME IMPORTANT SCIENTIFIC WORK IS TAKING PLACE. SO AS NOT TO OBSTRUCT THE VIEW OF THE ALPS, THIS WORK IS HAPPENING UNDERGROUND. CERN, THE EUROPEAN AGENCY THAT DOES RESEARCH IN SUB ATOMIC PHYSICS HAS RECENTLY LAUNCHED THE LARGE HADRON COLLIDER. THIS CIRCULAR TUNNEL WILL ACCELERATE SUBATOMIC PARTICLES TO NEARLY THE SPEED OF LIGHT AND HAVE THEM COLLIDE INTO EACH OTHER. TO UNDERSTAND WHAT A SUBATOMIC PARTICLE IS TRANSCRIPT—Geometry Applications: Points and Lines 3 LET'S START WITH AN ATOM. AN ATOM IS ONE OF THE SMALLEST PARTICLES THAT CAN STILL BE CALLED A SUBSTANCE. GOLD CAN EXIST AS AN ATOM BUT THERE IS NO SUBATOMIC VERSION OF GOLD. AN ATOM INCLUDES ELECTRONS, NEUTRONS AND PROTONS. LET'S LOOK AT THE SIMPLEST ATOM, HYDROGEN, WHICH CONSISTS OF A PROTON AND AN ELECTRON. THE PROTON IS A SUBATOMIC PARTICLE WHOSE SIZE IS IN THE NEIGHBORHOOD OF 10 TO THE -13TH METERS, WHICH IS INFINITESIMALLY SMALL. IN TERMS OF GEOMETRY, IS IT POSSIBLE TO CONSIDER A PROTON A GEOMETRIC POINT? AFTER ALL, IT IS SO SMALL THAT IT CAN'T BE SEEN BY THE HUMAN EYE. WHAT WE NEED IS A MATHEMATICAL DEFINITION OF A GEOMETRIC POINT. WE CAN USE EUCLID'S DEFINITION. WHAT THIS MEANS IS THAT A POINT HAS NO SIZE OR DIMENSION. HOWEVER SMALL A SUBATOMIC PARTICLE IS, IT IS STILL A MEASUREMENT IN SPACE. A GEOMETRIC POINT HAS NO SIZE BUT SIMPLY A LOCATION IN SPACE. TRANSCRIPT—Geometry Applications: Points and Lines 4 WE CAN USE THIS CONCEPT OF A LOCATION IN SPACE TO HELP EXPLAIN WHAT HAPPENS WITH SUBATOMIC PARTICLES. HERE ARE TWO POINTS LABELED "A" AND "B". SUPPOSE THAT EACH REPRESENTS THE LOCATION OF A SUBATOMIC PARTICLE IN THE LARGE HADRON COLLIDER AND SUPPOSE THESE PARTICLES ARE MOVING TOWARD EACH OTHER. THERE ARE THREE POSSIBLE OUTCOMES. IN ONE OUTCOME, A AND B MOVE PAST EACH OTHER WITH A MOVING ABOVE B. IN ANOTHER OUTCOME, A AND B MOVE PAST EACH OTHER WITH A MOVING BELOW B. IN THE THIRD SCENARIO, THE ONE WE'LL BE EXPLORING, A AND B COLLIDE. IN THE CASE OF THE SUBATOMIC PARTICLES, WHEN THEY COLLIDE SOMETIMES SPARKS FLY. LET'S SEE WHAT THIS MEANS GEOMETRICALLY BY USING THE TI-NSPIRE. TURN ON THE TI-NSPIRE. CREATE A NEW DOCUMENT. YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT. CREATE A GRAPHS AND GEOMETRY WINDOW. SELECT THE POINT TOOL. CLICK ON MENU, AND UNDER "POINTS AND LINES" SELECT POINT. TRANSCRIPT—Geometry Applications: Points and Lines 5 MOVE THE POINTER TO THE MIDDLE PART OF THE SCREEN AND PRESS CLICK TO CREATE A POINT. MOVE THE POINTER TO A DIFFERENT PART OF THE SCREEN AND CREATE A SECOND POINT. NOW LABEL EACH POINT. PRESS ESCAPE AND MOVE THE CURSOR ABOVE THE FIRST POINT. PRESS AND HOLD THE CLICK KEY. PRESS CONTROL AND MENU AND SELECT THE LABEL OPTION. PRESS THE CAPS KEY AND THE LETTER A TO LABEL THE POINT. REPEAT THE LABELING PROCESS WITH THE OTHER POINT. LABEL IT B THEN PRESS ESCAPE. MOVE THE CURSOR OVER POINT A. PRESS AND HOLD THE CLICK KEY TO GRAB THE POINT. MOVE THE POINT SO THAT IT OVERLAPS POINT B. WHEN YOU PLACE ONE POINT OVER ANOTHER THIS WAY, WHAT IS HAPPENING GEOMETRICALLY? THE POINTS INTERSECT AND SHARE THE SAME LOCATION IN SPACE. REMEMBER THAT POINTS HAVE NO DIMENSION SO IT'S NOT AS IF ONE POINT IS CROWDING OUT ANOTHER. POINTS ARE NOT LIKE PARTICLES IN SPACE. IF THE POINTS SHARE THE SAME POSITION IN SPACE THEN THE DISTANCE BETWEEN POINTS A AND B IS ZERO. TRANSCRIPT—Geometry Applications: Points and Lines 6 NOW POSITION THE POINTS SO THAT THERE IS SOME DISTANCE BETWEEN THEM. LET A REPRESENT A SUBATOMIC PARTICLE MOVING IN THE DIRECTION OF B AND LET B REPRESENT A SUBATOMIC PARTICLE MOVING IN THE DIRECTION OF A. SINCE THE DISTANCE BETWEEN A AND B IS GREATER THAN ZERO, THEN THERE IS A THIRD POINT, C, MIDWAY BETWEEN A AND B. OTHERWISE THE DISTANCE BETWEEN A AND B WOULD BE ZERO. CONTINUING WITH THIS, THERE IS A POINT BETWEEN A AND C AND BETWEEN C AND D MIDWAY BETWEEN THOSE POINTS. THIS PROCESS CAN CONTINUE INFINITELY. WHY? BECAUSE A POINT REPRESENTS A POSITION IN SPACE, NOT AN AMOUNT OF SPACE. THERE ARE AN INFINITE NUMBER OF POINTS BETWEEN A AND B. THIS INFINITE NUMBER OF POINTS BETWEEN A AND B REPRESENTING THE PATH THAT THE TWOSUBATOMIC PARTICLES HAVE THAT COLLIDE WITH EACH OTHER GIVE RISE TO ANOTHER GEOMETRIC FORM: THE LINE. WE CAN USE EUCLID'S DEFINITION OF A LINE. A LINE IS BREADTHLESS LENGTH. WHAT THIS MEANS IS THAT A LINE HAS A LENGTH THAT CAN BE MEASURED BUT NOT A WIDTH OR HEIGHT. TRANSCRIPT—Geometry Applications: Points and Lines 7 BECAUSE IT IS MADE UP OF DIMENSIONLESS POINTS IT BECOMES A ONE DIMENSIONAL FIGURE. SO TWO SUBATOMIC PARTICLES THAT COLLIDE WITH EACH OTHER ARE MODELED BY TWO COLLINEAR POINTS, OR POINTS THAT ARE ON THE SAME LINE. TO CONSTRUCT A LINE, PRESS MENU AND UNDER "POINTS AND LINES", SELECT LINE. MOVE THE CURSOR ABOVE POINT A UNTIL THE POINTER TURNS INTO A POINTING HAND. PRESS ENTER. THEN MOVE THE POINTER TOWARDS POINT B. NOTICE HOW A LINE FOLLOWS THE MOVEMENT OF THE POINTER. WHEN THE POINTER IS ABOVE B PRESS ENTER AGAIN. YOU NOW HAVE THE LINE CONNECTING POINTS A AND B. IS THERE MORE THAN ONE LINE THAT CAN INTERSECT BOTH POINTS A AND B? USE THE LINE TOOL TO CONSTRUCT OTHER LINES THAT CONTAIN AT LEAST ONE OF THE POINTS. YOU CAN EASILY CREATE MANY LINES THAT CONTAIN ONE OF THE POINTS. YOU CAN EVEN CONSTRUCT LINES THAT HAVE NONE OF THE POINTS. BUT THERE IS ONLY ONE LINE THAT CROSSES THE TWO POINTS. WE CAN GENERALIZE THIS TO SAY THAT FOR ANY TWO POINTS TRANSCRIPT—Geometry Applications: Points and Lines 8 THERE IS A UNIQUE LINE THAT CROSSES THE TWO POINTS. IN OTHER WORDS, ANY TWO POINTS ARE COLLINEAR. CONSTRUCT A NEW LINE. FIND A CLEAR PART OF THE SCREEN. PRESS THE CLICK KEY ONCE. THEN MOVE THE POINTER TO ANOTHER PART OF THE SCREEN. PRESS THE CLICK KEY AGAIN TO COMPLETE THE LINE. NOTICE THAT YOU ARE IN "POINT ON" MODE. IN THIS MODE ANY POINTS THAT YOU ADD TO THE LINE ARE COLLINEAR. ADD SEVERAL MORE COLLINEAR POINTS. NOW ADD A POINT THAT ISN'T ON THE LINE. THINK OF THE SITUATION OF THREE POINTS. HOW CAN YOU ENSURE THAT THEY ARE COLLINEAR? CLEAR YOUR GEOMETRY WINDOW. PRESS MENU AND UNDER ACTIONS SELECT "DELETE ALL". CLICK OKAY TO CONFIRM THE DELETION. SELECT THE POINT TOOL. PRESS MENU, AND UNDER "POINTS AND LINES" SELECT POINT. PLACE THREE POINTS ON SCREEN THAT ARE CLEARLY NOT COLLINEAR. YOU KNOW THAT FOR ANY TWO POINTS THERE IS A UNIQUE LINE THAT INCLUDES THE TWO POINTS. BUT WHAT IS THE UNIQUE LINE THAT CAN INCLUDE ALL THREE? TRANSCRIPT—Geometry Applications: Points and Lines 9 SINCE THE THREE POINTS ARE NOT COLLINEAR THIS ISN'T POSSIBLE, BUT THERE IS A WAY OF MAKING THEM COLLINEAR. ACCESS THE LINE TOOL. CLICK ON ONE OF THE THREE POINTS AND CREATE A LINE TO ANOTHER ONE OF THE POINTS. THE TWO COLLINEAR POINTS ARE ON THE LINE AND THE THIRD POINT IS NOT ON THE LINE. TO MAKE THE THREE POINTS COLLINEAR HIGHLIGHT THE THIRD POINT AND PLACE IT ON THE LINE. A STREAM OF SUBATOMIC PARTICLES ARE MOVING TOWARD EACH OTHER. THOSE THAT COLLIDE WITH EACH OTHER CAN BE MODELED BY COLLINEAR POINTS. OR THEY CAN BE MODELED BY INTERSECTING LINES. TAKE A LOOK AT POINTS A AND B. WE KNOW THAT THERE IS A UNIQUE LINE THAT INCLUDES BOTH POINTS. BUT THERE ARE ALSO AN INFINITE NUMBER OF LINES THAT INCLUDE ONE OF THE POINTS. CLEAR YOUR GEOMETRY WINDOW. PRESS MENU AND UNDER ACTIONS SELECT "DELETE ALL". CLICK OKAY TO CONFIRM THE DELETION. SELECT THE POINT TOOL. PRESS MENU, AND UNDER "POINTS AND LINES" SELECT POINT. TRANSCRIPT—Geometry Applications: Points and Lines 10 PLACE TWO POINTS ON SCREEN. USE THE LINE TOOL TO CREATE TWO LINES IN SUCH A WAY THAT EACH LINE HAS ONE OF THE TWO POINTS. MAKE SURE THAT THE LINES INTERSECT. ACTIVATE THE LINE TOOL. MOVE THE POINTER TO WHERE THE LINES INTERSECT.
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