Title: Geometry Applications: Chords, Secants, and Segments

Title: Geometry Applications: Chords, Secants, and Segments

[Music]

[Music]

JUST A SHORT DISTANCE FROM THE ROMAN COLOSSEUM

IS AN EVEN OLDER STRUCTURE

THAT WAS JUST AS WELL KNOWN.

THE PANTHEON WAS ORIGINALLY BUILT

NEARLY A CENTURY BEFORE THE COLOSSEUM.

IT WAS A RELIGIOUS BUILDING MEANT TO HONOR

THE MANY GODS THE ROMANS WORSHIPPED.

ALTHOUGH IN LATER YEARS IT WAS CONVERTED INTO

A CHRISTIAN CHURCH.

FROM THE OUTSIDE YOU CAN SEE THE CIRCULAR DOME

AND THE BODY OF THE STRUCTURE LOOKS CYLINDRICAL.

BUT INSIDE THE CIRCULAR EXPANSE IS DRAMATIC.

THE DOMED STRUCTURE BECOMES A GRAND ARCHED CEILING.

WHAT ADDS TO THE EFFECT IS THAT THE INTERIOR

OF THE PANTHEON IS AS TALL AS IT IS WIDE.

IN OTHER WORDS, IMAGINE A GREAT CIRCLE

DEFINING THE INTERIOR OF THE SPACE.

AT THE TOP OF THE DOME IS AN OPENING

CALLED THE OCULUS, WHICH LETS IN SUNLIGHT.

IT IS THE ONLY SOURCE OF LIGHT FOR THE PANTHEON AND

ADDS A DRAMATIC LIGHTING EFFECT TO THE SPACE.

THE OCULUS LETS IN LIGHT AT DIFFERENT ANGLES

DEPENDING ON THE TIME OF DAY AND THE TIME OF YEAR.

AS LIGHT COMES IN THROUGH THE OCULUS

IT IS DISPERSED AND THE PATTERN OF LIGHT FORMED

CAN BE MODELED ON A CIRCLE WITH TWO CHORDS.

A CHORD IS A SEGMENT WHOSE ENDPOINTS ARE ON THE CIRCLE.

A DIAMETER OF A CIRCLE IS A SPECIAL KIND OF CHORD,

ONE THAT INTERSECTS THE CENTER OF THE CIRCLE.

TO SIMPLIFY, LET ONE POINT BECOME INTO TWO CHORDS

THAT REPRESENT THE SPAN OF LIGHT.

SUCH A PAIR OF CHORDS FORM AN ANGLE

CALLED AN INSCRIBED ANGLE.

THE INSCRIBED ANGLE MARKS OUT A PORTION OF THE CIRCLE

CALLED AN INTERCEPTED ARC.

THROUGHOUT THE YEAR THE OCULUS ALLOWS A PORTION

OF LIGHT THAT DEFINES AN INTERCEPTED ARC.

THE INSCRIBED ANGLE IS DIFFERENT FROM A

CENTRAL ANGLE WHICH IS MADE UP OF TWO RADII.

BUT THERE IS A RELATIONSHIP BETWEEN

AN INSCRIBED ANGLE AND A CENTRAL ANGLE WHICH

REVEALS SOME INTERESTING FEATURES OF THE PANTHEON.

LET'S EXPLORE THIS RELATIONSHIP ON THE NSPIRE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW.

CONSTRUCT A HORIZONTAL LINE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

MOVE THE POINTER TO THE LOWER LEFT-HAND

PART OF THE SCREEN.

PRESS ENTER.

PRESS THE RIGHT ARROW

TO CONSTRUCT THE HORIZONTAL LINE.

STOP PRESSING THE RIGHT ARROW

WHEN YOU REACH THE OTHER END OF THE SCREEN.

PRESS ENTER AGAIN.

THIS LINE REPRESENTS THE FLOOR OF THE PANTHEON.

LET'S NOW CONSTRUCT THE GREAT CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN

ABOVE THE HORIZONTAL LINE.

PRESS ENTER.

THIS DEFINES THE CENTER OF THE CIRCLE.

MOVE THE POINTER TOWARD THE HORIZONTAL LINE

SO THAT THE CIRCLE INTERSECTS THE LINE.

YOU'LL SEE AN ONSCREEN LABEL THAT SAYS "POINT ON".

PRESS ENTER.

THIS CIRCLE REPRESENTS THE GREAT CIRCLE

THAT MAKES UP THE INTERIOR OF THE PANTHEON.

THE TOP HALF OF THE CIRCLE REPRESENTS THE DOME

WHILE THE BOTTOM HALF OF THE CIRCLE

IS A VIRTUAL SEMI-CIRCLE.

NEXT, CONSTRUCT TWO CHORDS THAT INTERSECT

AT THE TOP OF THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

MOVE THE POINTER TO THE TOP OF THE CIRCLE

DIRECTLY ABOVE THE CENTER.

PRESS ENTER.

THEN MOVE THE POINTER TO ROUGHLY 5 O'CLOCK

ON THE CIRCLE AND PRESS ENTER AGAIN.

GO BACK TO THE TOP OF THE CIRCLE AND HAVE THE POINTER

HOVER OVER THE FIRST ENDPOINT OF THE SEGMENT.

PRESS ENTER AGAIN TO CREATE THE STARTING POINT

OF THE NEXT SEGMENT.

MOVE THE POINTER TO ROUGHLY 3 O'CLOCK AND PRESS ENTER.

YOU'VE NOW CREATED TWO CHORDS

THAT INTERSECT AT ONE ENDPOINT.

THEY DEFINE AN INTERCEPTED ARC AS SHOWN HERE.

THESE CHORDS REPRESENT THE LIGHT

POURING THROUGH THE OCULUS OF THE PANTHEON.

NOW CONSTRUCT THE CENTRAL ANGLE.

CONTINUING WITH THE SEGMENT TOOL,

MOVE THE POINTER SO THAT IT HOVERS OVER THE CENTER.

PRESS ENTER.

MOVE THE POINTER TO THE ENDPOINT

OF THE SECOND CHORD.

PRESS ENTER.

RETURN THE POINTER TO THE CENTER OF THE CIRCLE.

PRESS ENTER.

THEN MOVE THE POINTER TO THE ENDPOINT

OF THE FIRST CHORD.

PRESS ENTER ONCE MORE.

NOW MEASURE THE INSCRIBED ANGLE AND THE CENTRAL ANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

TO MEASURE AN ANGLE, SELECT THREE POINTS

THAT DEFINE THE ANGLE.

MOVE THE POINTER ABOVE ONE OF THE ENDPOINTS

OF THE CHORD ON THE CIRCLE.

PRESS ENTER.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE.

PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER TO THE ENDPOINT

OF THE OTHER CHORD.

PRESS ENTER ONE MORE TIME.

YOU'LL SEE AN ANGLE MEASUREMENT APPEAR

NEXT TO THE VERTEX OF THE ANGLE BEING MEASURED.

REPEAT THIS PROCESS FOR THE INSCRIBED ANGLE.

YOU'LL NOTICE THAT THE CENTRAL ANGLE IS

TWICE THE MEASURE OF THE INSCRIBED ANGLE.

EVEN IF YOU CHANGE THE SIZE OF THE INTERCEPTED ARC,

THE RELATIONSHIP BETWEEN TWO ANGLES STAYS THE SAME.

NOTICE THAT THE POSITION OF THE OCULUS IS AT THE

EXACT OPPOSITE FROM WHERE SOMEONE WOULD BE STANDING

IN THE CENTER OF THE PANTHEON.

FROM THAT POINT OF VIEW, THE INSCRIBED ANGLE

IS IDENTICAL TO THAT OF THE OCULUS.

CREATE AN ANGLE USING THE SEGMENT TOOL

AND MEASURE IT TO VERIFY THIS.

SO THE GEOMETRY OF THE PANTHEON IS SUCH THAT

SOMEONE STANDING IN THE CENTER OF THE FLOOR

WOULD HAVE THE SAME VIEW OF THE LIGHT ON THE WALLS

AS THE OCULUS.

THIS IS IMPORTANT SINCE THE OCULUS WAS MORE THAN

JUST A HOLE IN THE ROOF.

THE OCULUS WAS MEANT TO SYMBOLIZE THE SUN,

THE SOURCE OF ALL LIGHT AND POWER.

SO THE PANTHEON WAS MEANT TO GIVE SOMEONE

THE VIEW FROM THE SUN.

IN A WAY, THE PANTHEON WAS MEANT TO ELEVATE

AND EXALT THE OBSERVER AND GIVE THEM A SENSE

OF THE POWER OF THE UNIVERSE AROUND THEM.

BUT THE OCULUS IS NOT A PIN PRICK OF LIGHT

THAT LETS ONLY A SMALL AMOUNT OF LIGHT IN.

IN FACT IT IS A CIRCLE WITH A DIAMETER OF 27 FEET.

WHILE THIS IS STILL A SMALL PORTION OF THE

OVERALL DIAMETER OF 142 FEET FOR THE GREAT CIRCLE

OF THE PANTHEON, IT IS STILL A SIZABLE HOLE.

SO THE OCULUS LETS IN A COLUMN OF LIGHT

WHICH LEADS TO A CIRCULAR SPOT OF LIGHT

ON THE WALLS OF THE PANTHEON.

LET'S INVESTIGATE THE INTERCEPTED ARC

FORMED BY THIS COLUMN OF LIGHT ON THE NSPIRE.

CREATE A NEW GEOMETRY WINDOW.

PRESS THE HOME KEY AND SELECT GEOMETRY.

CREATE A CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO THE CENTER OF THE SCREEN.

PRESS ENTER TO DEFINE THE CENTER OF THE CIRCLE.

MOVE THE POINTER AWAY FROM THE CENTER

TO DEFINE A CIRCLE ABOUT HALF THE SIZE OF THE SCREEN.

PRESS ENTER AGAIN.

USE THE SEGMENT TOOL TO CREATE AND MEASURE

THE RADIUS OF THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

THEN MOVE THE POINTER TO THE CIRCLE.

PRESS ENTER.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE.

PRESS ENTER AGAIN.

NOW MEASURE THE RADIUS.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER OVER THE RADIUS AND PRESS ENTER.

MOVE THE POINTER TO THE SIDE OF THE RADIUS

AND PRESS ENTER AGAIN

TO PASTE THE MEASUREMENT ONSCREEN.

WE WANT THIS CIRCLE TO MODEL THE GREAT CIRCLE

OF THE PANTHEON.

THE RADIUS OF THAT CIRCLE IS 71 FEET.

USING A 10:1 SCALE,

CHANGE THE RADIUS OF THE CIRCLE TO 7.1.

PRESS ESCAPE AND HOVER OVER THE CIRCLE.

PRESS AND HOLD THE CLICK KEY TO GRASP THE CIRCLE.

USE THE NAVIGATION ARROWS TO RESIZE THE CIRCLE.

TRY TO GET THE SIZE OF THE RADIUS

AS CLOSE TO 7.1 AS POSSIBLE.

PRESS ENTER.

LOCK THIS VALUE.

MOVE THE POINTER OVER THE MEASUREMENT VALUE.

PRESS CONTROL AND MENU

AND SELECT THE ATTRIBUTES OPTION.

USE THE DOWN ARROW TO HIGHLIGHT THE ICON

THAT LOOKS LIKE AN OPEN LOCK.

USE THE RIGHT ARROW TO CHANGE IT TO A CLOSED LOCK.

PRESS ENTER.

THIS CIRCLE IS NOW A SCALE MODEL OF

THE GREAT CIRCLE OF THE PANTHEON.

SINCE THE MEASUREMENT IS LOCKED,

THE CIRCLE CANNOT BE RESIZED.

NOW HIDE THE RADIUS.

MOVE THE POINTER OVER THE RADIUS,

PRESS CONTROL AND MENU AND SELECT THE HIDE OPTION.

NOW USE THE SEGMENT TOOL TO MODEL THE OCULUS.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

MOVE THE POINTER NEAR THE TOP OF THE CIRCLE

AND PRESS ENTER TO PLACE A POINT ON THE CIRCLE.

NEXT, MOVE THE POINTER TO ANOTHER PART OF THE CIRCLE

MAKING SURE THE SEGMENT REMAINS

AS CLOSE TO HORIZONTAL AS POSSIBLE.

PRESS ENTER.

NOW MEASURE THE SEGMENT.

PRESS MENU AND UNDER MEASUREMENT

SELECT LENGTH.

MOVE THE POINTER OVER THE SEGMENT

AND PRESS ENTER ONCE TO RECORD THE MEASUREMENT.

MOVE THE POINTER AND PRESS ENTER

TO PLACE THE MEASUREMENT ON SCREEN.

PRESS ESCAPE AND MOVE THE POINTER

OVER ONE OF THE SEGMENTS.

PRESS AND HOLD THE CLICK KEY TO HIGHLIGHT THE POINT.

RE-SIZE THE SEGMENT SO THAT IT IS AS CLOSE TO 2.7

AS POSSIBLE.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

LOCK THE VALUE OF THE SEGMENT MEASURE.

MOVE THE POINTER OVER THE MEASUREMENT VALUE,

PRESS CONTROL AND MENU.

SELECT ATTRIBUTES AND CHANGE THE OPEN LOCK

TO A CLOSED LOCK.

PRESS ENTER.

NOW CONSTRUCT A SECANT FROM ONE OF THE ENDPOINTS

OF THE SEGMENT TO THE OPPOSITE END OF THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

MOVE THE POINTER TO ONE OF THE ENDPOINTS

OF THE SEGMENT AND PRESS ENTER.

MOVE THE POINTER TO THE OPPOSITE SIDE OF THE CIRCLE

AND PRESS ENTER AGAIN.

NEXT, CREATE ANOTHER SECANT

PARALLEL TO THE FIRST.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PARALLEL.

MOVE THE POINTER OVER THE SECANT AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE SECOND ENDPOINT

OF THE SEGMENT AT TOP OF THE CIRCLE.

PRESS ENTER AGAIN.

EXTEND THE SECOND SECANT.

PRESS ESCAPE.

MOVE THE POINTER TO THE END OF THE CIRCLE.

PRESS AND HOLD THE CLICK KEY

AND EXTEND THE LINE BEYOND THE CIRCLE.

THEN ADD AN INTERSECTION POINT

WHERE THE SECOND SECANT INTERSECTS THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER OVER THE CIRCLE AND PRESS ENTER.

THEN MOVE THE POINTER OVER THE SECANT

AND PRESS ENTER AGAIN.

YOU NOW HAVE A MODEL OF THE INSIDE OF THE PANTHEON

AND THE COLUMN OF LIGHT

THAT SHINES THROUGH THE OCULUS.

MEASURE THE INTERCEPTED ARC.

FIRST DEFINE THE INTERCEPTED ARC

USING THE CIRCLE ARC TOOL.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT CIRCLE ARC.

MOVE THE POINTER ABOVE ONE OF THE ENDPOINTS

OF THE ARC AND PRESS ENTER.

NEXT, MOVE THE POINTER TO THE MIDDLE OF THIS ARC,

MAKING SURE THAT THE NEW POINT YOU ADD

IS ON THE CIRCLE.

AND PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER TO THE OTHER ENDPOINT

OF THE ARC AND PRESS ENTER ONE MORE TIME.

MEASURE THE ARC.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE ARC AND

MAKE SURE THE "CIRCLE ARC" ONSCREEN TEXT APPEARS.

PRESS ENTER.

THE MEASUREMENT OF THE ARC IS A DISTANCE MEASUREMENT

BUT IT DOES CORRESPOND TO AN ANGLE MEASUREMENT.

THE RELATIONSHIP BETWEEN ARC LENGTH

AND THE ANGLE MEASURE OF THE INTERCEPTED ARC

IS S = R THETA WHERE S IS THE ARC LENGTH,

R IS THE RADIUS OF THE CIRCLE,

AND THE THETA IS THE ANGLE MEASURE

REPRESENTED BY THE INTERCEPTED ARC.

SOLVING FOR THETA WE GET THETA = S OVER R.

THIS GIVES THE ANGLE MEASURE IN A UNIT CALLED RADIANS.

TO CHANGE IT TO A DEGREE MEASURE,

MULTIPLY THIS EXPRESSION BY 360 OVER 2 PI.

SO THE ANGLE FORMULA BECOMES

THETA = 180S OVER PI R.

CREATE A FORMULA TO CONVERT THE ARC LENGTH TO AN ANGLE.

PRESS MENU AND UNDER ACTIONS SELECT TEXT.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN.

PRESS ENTER.

INPUT THE FORMULA 180S OVER PI R.

TO INPUT THE SYMBOL FOR PI, PRESS THE LIBRARY BUTTON

WHICH LOOKS LIKE AN OPEN BOOK.

PRESS 3 TO BRING UP THE SYMBOL PALLET AND SELECT PI.

POINT THE FORMULA TO THE VALUES ON THE CIRCLE.

PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.

MOVE THE POINTER ABOVE THE FORMULA.

PRESS ENTER.

MOVE THE POINTER ABOVE THE VALUE FOR THE RADIUS.

PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE VALUE FOR THE ARC LENGTH

AND PRESS ENTER AGAIN.

YOU'LL SEE THE VALUE FOR THE ANGLE MEASURE

OF THE ARC LENGTH.

MOVE THE POINTER NEXT TO THE FORMULA

AND PRESS ENTER TO PLACE THE VALUE ONSCREEN.

THE ANGLE MEASURE SHOULD BE ABOUT 22 DEGREES.

IN FACT, THE ANGLE MEASURE STAYS AT 22 DEGREES

NO MATTER HOW THE COLUMN OF LIGHT IS ORIENTED.

AS YOU MOVE THE PARALLEL LINES

TO DIFFERENT POSITIONS, MAKE SURE TO ALSO

MOVE THE MIDDLE POINT OF THE ARC.

THIS CONSTANT 22 DEGREE PATCH OF LIGHT SHINES

THROUGHOUT THE YEAR, BUT TAKES ON MORE SIGNIFICANCE

DURING FOUR KEY DAYS

THE WINTER AND SUMMER SOLSTICES

AND THE FALL AND SPRING EQUINOXES.

DURING THESE DAYS THE SPOT OF LIGHT

SHINES OVER THE ENTRANCE TO THE PANTHEON.

AS YOU HAVE SEEN, THESE FOUR DATES WERE

VERY SIGNIFICANT TO ANCIENT CULTURES.

SOMEONE ENTERING THE PANTHEON ON THESE DATES

WOULD BE BATHED IN LIGHT, AND ALL FOUR DATES

OCCUR ON OR ABOUT THE 22ND DAY OF MAY, JUNE,

SEPTEMBER AND DECEMBER.

THUS THE ANGLE MEASURE OF THE INTERCEPTED ARC

TAKES ON ADDITIONAL SIGNIFICANCE.

SO THE GEOMETRY OF THE PANTHEON

WAS VERY SIGNIFICANT FOR THE ROMANS.

THEY CREATED A STRUCTURE

THAT HAS LASTED FOR MILLENNIA

AND UTILIZES THE TIMELESS GEOMETRY OF CIRCLES.