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.
NOTICE THE ON-SCREEN MESSAGE THAT SAYS
"INTERSECTION POINT". PRESS ENTER.
YOU NOW HAVE A THIRD POINT WHERE THE LINES INTERSECT.
THE THREE POINTS ARE NOT COLLINEAR SINCE
THERE ISN'T ONE LINE THAT CONTAINS ALL THE POINTS.
BECAUSE A GEOMETRIC POINT IS A LOCATION IN SPACE,
THIS PROPERTY ALLOWS US TO MODEL
THE LOCATIONS OF SUB ATOMIC PARTICLES.
WHEN YOU KNOW THE LOCATION OF AN OBJECT
YOU CAN USE COORDINATES TO TRACK POSITION.
ASSIGNING COORDINATES TO POINTS
IS AN EXAMPLE OF COORDINATE GEOMETRY.
LET'S LOOK AT AN EXAMPLE ON THE INSPIRE.
PRESS THE HOME KEY AND CREATE A NEW
GRAPHS AND GEOMETRY WINDOW.
NOTICE THE COORDINATE GRID THAT APPEARS.
DRAW TWO POINTS ON THE GRID. PRESS MENU,
AND UNDER "POINTS AND LINES" SELECT POINT. TRANSCRIPT—Geometry Applications: Points and Lines 11
MOVE THE POINTER TO THE MIDDLE OF THE SCREEN
AND PRESS ENTER TO CREATE THE FIRST POINT.
MOVE THE POINTER TO A DIFFERENT PART
OF THE SCREEN. PRESS ENTER AGAIN.
YOU NOW HAVE TWO POINTS.
EACH OF THESE POINTS HAS A POSITION IN SPACE
AND HAS COORDINATES ASSIGNED TO THEM
IN THIS COORDINATE SYSTEM.
TO SEE THE COORDINATES PRESS MENU AND UNDER ACTION
SELECT "COORDINATES AND EQUATIONS".
USE THE NAV PAD TO MOVE THE POINTER
ABOVE ONE OF THE POINTS.
YOU'LL SEE COORDINATES APPEAR NEXT TO THE POINT.
PRESS ENTER TWICE TO PASTE THE COORDINATES ON SCREEN.
REPEAT WITH THE OTHER POINT.
YOU CAN ALSO ASSIGN DIFFERENT COORDINATES
TO THESE POINTS.
WHEN YOU DO SO, THE POINTS WILL MOVE
TO THE APPROPRIATE LOCATIONS.
PRESS THE ESCAPE BUTTON.
NOW MOVE THE POINTER ABOVE THE X COORDINATE
OF ONE OF THE POINTS.
PRESS ENTER TWICE TO SEE A CURSOR APPEAR
WITHIN THE TEXT FIELD OF THE COORDINATE. TRANSCRIPT—Geometry Applications: Points and Lines 12
PRESS THE CLEAR BUTTON SEVERAL TIMES
TO DELETE THE COORDINATE.
REPLACE THE COORDINATE WITH THE NUMBER 2.
PRESS ENTER.
NOTICE THAT THE POINT SHIFTS HORIZONTALLY
SINCE YOU CHANGED THE X COORDINATE.
REPEAT THIS PROCESS FOR THE REMAINING COORDINATES.
CHANGE THE Y COORDINATE OF THE FIRST POINT TO 3
AND CHANGE THE X,Y COORDINATES
OF THE SECOND POINT TO -2, -4.
TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.
GEOMETRIC POINTS DO NOT HAVE SIZE.
THEY ARE LOCATIONS IN SPACE.
GEOMETRIC SPACE IS MADE UP OF ALL GEOMETRIC POINTS.
BECAUSE A POINT HAS A LOCATION,
COORDINATES CAN BE ASSIGNED TO IT.
THE MOVEMENT OF AN OBJECT IN SPACE CAN BE THOUGHT OF
AS THE CHANGE IN LOCATION OF A SET OF GEOMETRIC POINTS. Geometry Applications: Points and Lines, Segment 3: Lines
SUNRISE OVER THE HOUSTON SKYLINE.
SEEN IN PROFILE, THE SKYLINE IS A JAGGED OUTLINE.
BUT SEEN FROM ABOVE, THE CITY REVEALS AN ORDERLY
ARRANGEMENT OF PARALLEL AND PERPENDICULAR LINES.
MANY CITIES ARE ARRANGED THIS WAY. TRANSCRIPT—Geometry Applications: Points and Lines 13
MANY CITIES HAVE BEEN ARRANGED THIS WAY
FOR CENTURIES.
WHAT PROPERTIES OF LINES MAKE THIS A PREFERRED WAY
OF ORGANIZING A CITY OR COMMUNITY?
IN THE PREVIOUS SECTION YOU LEARNED ABOUT LINES
IN THE CONTEXT OF POINTS.
RECALL EUCLID'S DEFINITION.
A LINE IS BREADTHLESS LENGTH.
A LINE IS MADE UP OF AN INFINITE NUMBER
OF COLLINEAR POINTS.
IN FACT, THE SHORTEST DISTANCE BETWEEN TWO POINTS
IS A LINE.
HOW DO WE KNOW THIS?
HERE ARE POINTS A AND B.
WE KNOW THAT THERE IS ONLY ONE LINE
THAT CONNECTS THESE POINTS.
BUT HOW DO WE KNOW THAT THE SHORTEST DISTANCE
BETWEEN THE POINTS IS DEFINED BY THIS LINE?
LET'S ASSUME THAT THE LINE ISN'T THE SHORTEST DISTANCE.
THEN IT FOLLOWS THAT THE SHORTEST DISTANCE
WOULD GO THROUGH ANOTHER PATH
AND CROSS ANOTHER POINT: C.
POINT C IS NOT ON THE LINE.
SO THE SHORTEST DISTANCE FROM A TO B TRANSCRIPT—Geometry Applications: Points and Lines 14
WOULD GO THROUGH C.
BUT THE LINE FROM A TO C CANNOT BE
THE SHORTEST PATH FROM A TO C
USING THE SAME LOGIC AS THE PATH FROM A TO B.
THERE MUST BE ANOTHER PATH
THAT CROSSES ANOTHER POINT, D,
NOT ON THE LINE CONNECTING A AND C.
BUT ONCE AGAIN THE LINE CONNECTING POINTS A AND D
CANNOT POSSIBLY BE THE SHORTEST DISTANCE
BETWEEN THESE TWO POINTS.
AS YOU CAN SEE, WE ARE CONTINUALLY HAVING TO ADD
NEW POINTS AND THIS WOULD CONTINUE AD INFINITUM.
NOT ONLY THAT...
NOTICE THAT AS EACH NEW POINT IS ADDED,
THE ORIGINAL DISTANCE FROM A TO B KEEPS INCREASING.
IN FACT, AS THE NUMBER OF POINTS APPROACHES INFINITY,
SO DOES THE DISTANCE BETWEEN A AND B.
SO USING LOGICAL REASONING WE CONCLUDE THAT
THE SHORTEST DISTANCE BETWEEN TWO POINTS IS A LINE.
ANY OTHER NON-COLLINEAR PATH WOULD BE A LONGER DISTANCE.
THIS HAS IMPORTANT IMPLICATIONS
FOR CITY PLANNING.
WHY? LET'S GO BACK TO HOUSTON.
TO TRAVEL FROM ONE PART OF THE CITY TO ANOTHER TRANSCRIPT—Geometry Applications: Points and Lines 15
INVOLVES WALKING A SERIES OF STRAIGHT LINES.
BUT SUPPOSE THE CITY WAS LAID OUT
ALONG A CIRCULAR GRID LIKE THIS.
IN SOME CASES GOING FROM POINT A TO B
WOULD BE SHORTER THAN A STRAIGHT PATH.
LET'S EXAMINE THIS ON 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.
CLICK ON MENU AND UNDER "POINTS AND LINES"
SELECT SEGMENT.
MOVE THE POINTER TO THE MIDDLE PART OF THE SCREEN
AND PRESS ENTER TO CREATE A POINT.
MOVE THE POINTER UP.
PRESS ENTER TO DEFINE THE FIRST LINE SEGMENT.
PRESS ENTER AGAIN ON TOP OF THE POINT YOU JUST CREATED.
MOVE THE POINTER TO THE RIGHT.
PRESS ENTER ONCE MORE TO DEFINE THE SECOND SEGMENT.
NOW LABEL EACH POINT.
PRESS ESCAPE AND MOVE THE CURSOR
ABOVE THE LAST POINT YOU CREATED.
PRESS CONTROL AND MENU AND SELECT THE LABEL OPTION. TRANSCRIPT—Geometry Applications: Points and Lines 16
PRESS THE CAPS KEY AND THE LETTER C TO LABEL THE POINT.
REPEAT THE LABELING PROCESS WITH THE OTHER POINT.
LABEL IT B.
REPEAT THE LABELING PROCESS WITH THE THIRD POINT.
LABEL IT A.
TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.
THEN PRESS ESCAPE.
WE KNOW THAT THE SHORTEST DISTANCE BETWEEN A AND B
IS THE LINE THAT PASSES THROUGH THEM
AND THAT THE SHORTEST DISTANCE BETWEEN B AND C
IS THE LINE CONNECTING THOSE POINTS.
SO IF THIS WERE A CITY BLOCK,
THIS WOULD BE THE QUICKEST WAY TO GO FROM A TO C.
NOW SUPPOSE THE CITY WAS LAID OUT IN A CIRCULAR GRID.
WE'LL USE THE ARC TOOL TO CONNECT POINTS A AND C.
PRESS MENU AND UNDER "POINTS AND LINES"
SELECT THE CIRCLE ARC TOOL.
MOVE THE POINTER ABOVE POINT A.
PRESS ENTER.
NOW MOVE THE POINTER MIDWAY BETWEEN A AND C.
PRESS ENTER AGAIN.
FINALLY MOVE THE POINTER TO POINT C.
PRESS ENTER ONE MORE TIME.
YOU'LL SEE A CURVED ARC FROM POINT A TO C. TRANSCRIPT—Geometry Applications: Points and Lines 17
DEPENDING ON THE SIZE OF THE ARC,
THE DISTANCE FROM A TO C CAN BE SHORTER OR LONGER
THAN THE STRAIGHT LINE DISTANCE FROM A TO C.
TO MEASURE THE LENGTHS, PRESS MENU
AND UNDER MEASUREMENT SELECT LENGTH.
MOVE THE POINTER ABOVE THE ARC.
PRESS ENTER TO SEE THE MEASUREMENT.
PRESS ENTER AGAIN TO PLACE THE MEASUREMENT
ON THE SCREEN.
REPEAT THIS PROCESS FOR THE LINE SEGMENTS.
TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.
AFTER YOU ARE DONE MEASURING THE LENGTHS,
PRESS ESCAPE.
MOVE THE POINTER ABOVE THE MIDDLE POINT ON THE ARC.
MOVE TO VARIOUS LOCATIONS ON SCREEN.
YOU'LL SEE THAT THE DISTANCE FROM A TO C
WILL VARY AND WILL BE SHORTER OR LONGER
THAN THE STRAIGHT LINE DISTANCES.
YOU CAN ALSO CREATE AN ON-SCREEN FORMULA
FOR CALCULATING THE CHANGING LENGTHS
OF THE STRAIGHT SIDES.
PRESS MENU AND UNDER ACTIONS SELECT TEXT.
MOVE THE POINTER AND PRESS THE CLICK KEY.
YOU SHOULD SEE THE TEXT CURSOR. TRANSCRIPT—Geometry Applications: Points and Lines 18
INPUT THE FORMULA AB+BC AND PRESS ENTER.
LINK THIS FORMULA TO THE VALUES
OF THE STRAIGHT SIDE LENGTHS.
PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.
MOVE THE POINTER ABOVE THE TEXT FORMULA
AND PRESS ENTER.
YOU WILL BE ASKED TO FIRST LINK AB AND THEN BC.
FOR AB MOVE THE POINTER TO THE LENGTH MEASUREMENT
OF SIDE AB.
PRESS ENTER.
REPEAT FOR SIDE BC.
WHEN YOU ARE DONE PRESS ESCAPE.
SO IT IS POSSIBLE TO CREATE A CIRCULAR STREET GRID
THAT HAS SHORTER DISTANCES BETWEEN TWO POINTS.
WHY AREN'T MORE CITIES ARRANGED IN A CIRCULAR GRID?
THE ANSWER HAS MORE TO DO WITH PHYSICS
THAN WITH GEOMETRY.
TRAVELING IN A STRAIGHT LINE TAKES LESS ENERGY
THAN TRAVELING ALONG A CURVE.
IF A CITY WERE LAID OUT IN A CIRCULAR GRID
ALL THE CARS AND TRUCKS TRAVELING ALONG THOSE ROADS
WOULD USE MUCH MORE FUEL THAN IF THOSE SAME VEHICLES
WERE TRAVELING ALONG A STRAIGHT LINE GRID.
A STRAIGHT LINE IS NOT ONLY TRANSCRIPT—Geometry Applications: Points and Lines 19
THE SHORTEST DISTANCE BETWEEN TWO POINTS,
IT IS ALSO THE MOST FUEL EFFICIENT.
THE SIMPLE ELEGANT LINES OF A STREET GRID
ALSO REVEAL AN UNDERLYING EFFICIENCY.
SINCE THE CENTER OF A CITY LIKE HOUSTON
IS ALSO THE CENTER OF COMMERCE WITH LOTS OF
PEOPLE AND VEHICLES MOVING BACK AND FORTH,
THEN THE MOST EFFECTIVE ARRANGEMENT
INVOLVES AN UNDERLYING STRAIGHT-LINE GRID.
BUT THERE ARE DIFFERENT WAYS OF ARRANGING
A STRAIGHT LINE GRID... LIKE THE ONE SHOWN HERE.
NOTICE THAT WITH THESE GRIDS
THE LINES ARE NOT ALL PARALLEL.
NOT ONLY DOES IT NOT LOOK AS ORDERLY
AS THE PREVIOUS GRIDS BUT IS THERE A DIFFERENT
TYPE OF INEFFICIENCY INTRODUCED
WITH THIS KIND OF ARRANGEMENT?
LET'S EXPLORE THE GEOMETRY OF PARALLEL LINES.
YOU KNOW THAT FOR ANY TWO POINTS A AND B
THERE IS A UNIQUE LINE THAT CROSSES THE TWO POINTS.
YOU ALSO KNOW THAT FOR ANY THREE
NON-COLLINEAR POINTS, A, B AND C,
THERE IS NO LINE THAT INCLUDES ALL THREE POINTS.
THERE IS, HOWEVER, A GEOMETRIC OBJECT TRANSCRIPT—Geometry Applications: Points and Lines 20
THAT INCLUDES THESE POINTS:
IT IS THE GEOMETRIC STRUCTURE KNOWN AS A PLANE.
A PLANE IS A TWO DIMENSIONAL FLAT SURFACE
MADE UP OF AN INFINITE NUMBER OF POINTS.
IT ALSO CONTAINS AN INFINITE NUMBER OF LINES.
THE PLANE EXTENDS INDEFINITELY
IN BOTH DIRECTIONS.
BECAUSE THE PLANE IS MADE OF POINTS IT HAS NO THICKNESS.
SO LET'S RESTRICT OURSELVES TO THIS PLANE.
SUPPOSE THERE ARE TWO LINES, L1 AND L2.
WE DEFINE PARALLEL LINES TO MEAN THAT
PARALLEL STRAIGHT LINES ARE STRAIGHT LINES WHICH,
BEING IN THE SAME PLANE AND BEING
PRODUCED INDEFINITELY IN BOTH DIRECTIONS,
DO NOT MEET ONE ANOTHER IN EITHER DIRECTION.
NO MATTER HOW FAR LINES L1 AND L2 EXTEND,
THE TWO LINES WILL NEVER INTERSECT.
YOU'LL SEE THAT A CITY GRID IS MADE UP OF
PARALLEL LINES.
WHAT ADVANTAGE DO PARALLEL LINES HAVE
OVER NON PARALLEL LINES?
LET'S EXPLORE ON THE TI-NSPIRE.
CLEAR THE PREVIOUS DOCUMENT YOU WERE USING
OR CREATE A NEW ONE. TRANSCRIPT—Geometry Applications: Points and Lines 21
TO CLEAR THE SCREEN PRESS MENU
AND UNDER ACTIONS SELECT "DELETE ALL".
PRESS ENTER TO SELECT OKAY.
CREATE A LINE.
PRESS MENU AND UNDER "POINTS AND LINES"
SELECT LINE.
MOVE THE POINTER TO THE MIDDLE LEFT
PART OF THE SCREEN AND PRESS ENTER.
MOVE THE POINTER TO THE RIGHT.
YOU'LL SEE THE LINE TAKING SHAPE.
PRESS ENTER WHEN THE POINTER IS ON THE
OTHER SIDE OF THE SCREEN.
NOW CREATE A PARALLEL LINE.
PRESS MENU AND UNDER CONSTRUCTION
SELECT PARALLEL.
MOVE THE POINTER DOWN ABOUT
A QUARTER OF THE LENGTH OF THE SCREEN.
PRESS ENTER.
THE SECOND LINE YOU'VE CREATED
IS PARALLEL TO THE FIRST.
IF YOU MANIPULATE THE FIRST LINE,
INCLUDING ROTATING IT,
THE SECOND LINE WILL REMAIN PARALLEL.
WE WILL MEASURE THE DISTANCE BETWEEN THE TWO LINES. TRANSCRIPT—Geometry Applications: Points and Lines 22
ADD TWO POINTS, ONE ON EACH LINE.
PRESS MENU AND UNDER "POINTS AND LINES"
SELECT SEGMENT.
MAKE SURE THE POINTER IS ON ONE OF THE LINES.
PRESS ENTER TO ADD ONE OF THE ENDPOINTS
OF THE LINE SEGMENT.
MOVE THE POINTER TO THE OTHER LINES.
PRESS ENTER AGAIN TO ADD THE OTHER ENDPOINT.
YOU WILL NOW CREATE MULTIPLE COPIES OF THIS SEGMENT
USING THE TRANSLATE FEATURE OF THE INSPIRE.
CLICK ON MENU AND UNDER TRANSFORMATIONS
SELECT TRANSLATE.
MOVE THE POINTER SO THAT IT HOVERS OVER
THE SEGMENT YOU'VE CREATED.
PRESS ENTER.
MOVE THE POINTER DOWN TO HOVER OVER THE BOTTOM POINT.
PRESS ENTER AGAIN.
NOW MOVE THE POINTER TO THE LEFT.
WHAT YOU'RE DOING IS TRANSLATING,
OR MOVING A COPY OF THE ORIGINAL SEGMENT
TO A NEW LOCATION.
PRESS ENTER.
REPEAT THIS PROCESS SEVERAL MORE TIMES
TO CREATE SEVERAL SEGMENTS. TRANSCRIPT—Geometry Applications: Points and Lines 23
TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.
NOW MEASURE THE LENGTH OF THE SEGMENT.
PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.
MOVE THE POINTER TO EACH OF THE SEGMENTS.
YOU'LL SEE THE MEASUREMENT OF THE SEGMENTS
APPEAR ON SCREEN.
YOU'LL NOTICE THAT EACH SEGMENT HAS THE SAME LENGTH.
PRESS ENTER TWICE OVER ONE OF THE SEGMENTS
TO RECORD THE LENGTH OF THE SEGMENT.
SO ONE OF THE PROPERTIES OF PARALLEL LINES
IS THAT THEY ARE THE SAME DISTANCE FROM EACH OTHER
THROUGHOUT THE EXPANSE OF THE LINES.
AS YOU CAN SEE FROM THE CITY GRID OF HOUSTON,
MANY OF THE STREETS ARE PARALLEL TO EACH OTHER.
NOW LET'S EXPLORE WHICH LINE SEGMENT IS THE SHORTEST.
FOR THIS WE WILL BE MEASURING THE ANGLE
FORMED BY THE SEGMENT IN ONE OF THE PARALLEL LINES.
PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.
YOU NEED THREE POINTS TO DEFINE A LINE.
USE THE NAV PAD TO MOVE THE POINTER TO THE MIDDLE OF THE
LINE SEGMENT THAT HAS THE LENGTH MEASUREMENT SHOWING.
PRESS ENTER.
MOVE THE POINTER DOWN TO THE POINT
THAT INTERSECTS THE PARALLEL LINE. TRANSCRIPT—Geometry Applications: Points and Lines 24
PRESS ENTER.
MOVE THE POINTER TO THE RIGHT OF THIS POINT
AND PRESS ENTER AGAIN.
THE ANGLE MEASURE APPEARS AND SHOULD BE 90 DEGREES.
PRESS ESCAPE.
GO TO ONE OF THE ENDPOINTS OF THE LINE SEGMENT.
SELECT THE POINT AND MOVE IT RIGHT AND LEFT.
NOTE HOW THE ANGLE MEASURE
AND THE SEGMENT LENGTH CHANGE.
FOR WHICH VALUES OF THE ANGLE
IS THE SEGMENT LENGTH THE LEAST?
90 DEGREES.
IN OTHER WORDS, A LINE THAT IS PERPENDICULAR
TO THE PARALLEL LINES IS THE SHORTEST DISTANCE.
SO NOW WE GET A BETTER UNDERSTANDING
OF WHY A CITY GRID IS MADE UP OF
PARALLEL AND PERPENDICULAR LINES.
THE USE OF PARALLEL LINES ENSURES THAT CITY BLOCKS
ARE EQUIDISTANT FROM EACH OTHER.
THE USE OF PERPENDICULAR LINES
ENSURES THE SHORTEST PATH FROM ONE BLOCK TO ANOTHER.
AND HERE IS THE RESULT.
LOOK AT THIS DIAGRAM OF A SECTION OF DOWNTOWN.
SUPPOSE YOU WANT TO GO FROM POINT A TO POINT B. TRANSCRIPT—Geometry Applications: Points and Lines 25
THERE ARE MULTIPLE WAYS OF GETTING FROM A TO B,
SOME OF WHICH ARE SHOWN HERE.
BECAUSE OF THE USE OF PARALLEL
AND PERPENDICULAR LINES
IT DOESN'T MATTER WHICH ROUTE YOU TAKE.
YOU TRAVEL THE SAME DISTANCE.
WHY IS THIS AN ADVANTAGE?
SINCE ANY PATH IS THE SAME DISTANCE,
THIS MEANS THAT TRAFFIC CAN BE EQUALLY DISTRIBUTED.
NO ONE PATH IS SHORTER THAN ANOTHER AND SO
TRAFFIC WON'T CLUSTER IN SOME AREAS OVER OTHERS.
THIS IS AN EFFICIENT WAY FOR PEOPLE IN VEHICLES
TO MOVE THROUGH THE DOWNTOWN AREA.
FINALLY, LET'S LOOK AT ANOTHER PROPERTY
OF PARALLEL LINES.
YOU SAW THAT THE SHORTEST DISTANCE
FROM ONE PARALLEL TO ANOTHER IS A LINE
PERPENDICULAR TO THE TWO PARALLEL LINES.
NOW LET'S INVESTIGATE WHAT HAPPENS WHEN
THE INTERSECTING LINE IS NOT PERPENDICULAR.
PRESS MENU AND UNDER "POINTS AND LINES"
SELECT THE LINE TOOL.
MOVE THE POINTER TO THE LOWER PARALLEL LINE
TO A PLACE WHERE THERE ISN'T ALREADY A POINT. TRANSCRIPT—Geometry Applications: Points and Lines 26
PRESS ENTER.
NOW MOVE THE POINTER TO THE OTHER PARALLEL LINE
AND MAKE SURE THAT THE LINE YOU CREATE
IS AT A SLANT LIKE THE ONE SHOWN.
PRESS ENTER.
WHAT YOU NOW HAVE ARE TWO PARALLEL LINES
CUT BY A TRANSVERSAL.
COMPARE THE ANGLE MEASURES
FOR THE HIGHLIGHTED ANGLE SHOWN.
USE THE ANGLE MEASURE TOOL.
PRESS MENU AND UNDER MEASUREMENTS SELECT ANGLE.
REMEMBER THAT WITH THE ANGLE MEASUREMENT TOOL
YOU NEED TO DEFINE THREE POINTS.
STARTING WITH THE FIRST ANGLE
IN THE UPPER RIGHT-HAND SECTION,
MOVE THE POINTER ABOVE THE TRANSVERSAL
AND PRESS ENTER.
MOVE THE POINTER TO THE INTERSECTION POINT
OF THE TRANSVERSAL AND ONE OF THE PARALLEL LINES
AND PRESS ENTER.
THEN MOVE THE POINTER ABOVE THE PARALLEL LINE.
REPEAT THIS FOR THE OTHER ANGLES.
PAUSE THE VIDEO TO MEASURE THE ANGLES.
WHEN YOU ARE DONE, PRESS THE ESCAPE KEY. TRANSCRIPT—Geometry Applications: Points and Lines 27
NOW SELECT ONE OF THE INTERSECTION POINTS
OF THE TRANSVERSAL AND ONE OF THE PARALLEL LINES.
MOVE IT LEFT OR RIGHT TO CHANGE THE ORIENTATION
OF THE LINE.
NOTICE THAT THESE PAIRS OF ANGLES REMAIN EQUAL,
OR CONGRUENT, TO EACH OTHER.
FURTHERMORE, THE OTHER PAIR OF ANGLES
ALSO REMAIN CONGRUENT.
FINALLY, THE SUM OF ADJACENT ANGLES IS 180 DEGREES.
AS WE LOOK ON THE HOUSTON SKYLINE
WE KNOW THAT ITS IRREGULAR CONTOURS
REVEAL AN ORDERED GEOMETRY AT ITS CORE.
THIS ORDER IS BASED ON THE PROPERTIES
OF PARALLEL AND PERPENDICULAR LINES.