Quadrilaterals--FullProgram.mp4

Geometry Applications: Quadrilaterals

[Music]

Title: Geometry Applications: Quadrilaterals

Title: Geometry Applications: Geometry Basics: Quadrilaterals

STONEHENGE IS ONE OF THE WORLD'S MOST FAMOUS

LANDMARKS AND IT IS MOSTLY THOUGHT OF

AS A CIRCULAR STRUCTURE.

BUT WHAT MAKES STONEHENGE POSSIBLE

IS THE POST AND LINTEL CONSTRUCTION

WHERE SLABS OF STONE ARE ARRANGED

HORIZONTALLY AND VERTICALLY.

THE MOST COMMON EXAMPLE OF

POST AND LINTEL CONSTRUCTION

IS THE FRAMEWORK AROUND THE DOORWAY IN YOUR HOUSE.

THE POSTS ARE THE VERTICAL SUPPORTS

AND THE LINTEL IS THE HORIZONTAL SPAN.

THE FLOOR COMPLETES THE QUADRILATERAL

THAT DEFINES THE ENTRYWAY

AND THE SHAPE IS USUALLY A RECTANGLE.

IN FACT, THE FRONT OF MOST HOMES IS ANOTHER

EXAMPLE OF A POST AND LINTEL STRUCTURE.

THE TRIANGULAR ROOF USUALLY SITS ON

A POST AND LINTEL BASE.

IN THE ANCIENT WORLD THIS SAME TYPE OF CONSTRUCTION

WAS USED WITH TEMPLES.

WE ARE QUITE AT HOME WITH QUADRILATERALS.

SQUARE AND RECTANGULAR FRAMES

ARE FOUND THROUGHOUT THE HOUSE.

IN FACT, MOST HOMES HAVE A FLOOR PLAN

THAT IS MADE UP OF RECTANGLES AND SQUARES.

WHILE THERE ARE SOME CIRCULAR SHAPED HOMES

AND SOME THAT ARE BASED ON POLYGONS,

THE OVERWHELMING MAJORITY USE QUADRILATERAL FORMS.

WHY ARE SQUARE AND RECTANGULAR FORMS

SO PREVALENT IN HOMES?

WHAT PROPERTIES OF THESE SHAPES MAKE THEM IDEAL

FOR THESE TYPES OF STRUCTURES?

IN THIS PROGRAM YOU WILL LEARN ABOUT

THE PROPERTIES OF QUADRILATERALS

IN ORDER TO SOLVE REAL WORLD PROBLEMS.

SPECIFICALLY, THIS PROGRAM WILL ADDRESS

THE FOLLOWING CONCEPTS:

IN THE COUNTRYSIDE OF WESTERN PENNSYLVANIA

IS ONE OF AMERICA'S MOST FAMOUS

ARCHITECTURAL LANDMARKS.

THIS HOUSE, KNOWN AS "FALLING WATER",

WAS DESIGNED BY FRANK LLOYD WRIGHT, ONE OF THE

GREATEST ARCHITECTS OF THE 20TH CENTURY.

ORIGINALLY BUILT AS A VACATION HOME

FOR A PROMINENT PITTSBURGH FAMILY,

"FALLING WATER" IS NOW A HISTORICAL LANDMARK

RECEIVING THOUSANDS OF TOURISTS EACH YEAR.

THIS DISTINCTIVE HOME SEEMS TO EMERGE

FROM THE UNDERLYING ROCKS AND TREES.

AND THIS IS WHAT FRANK LLOYD WRIGHT INTENDED.

"FALLING WATER" IS AN EXAMPLE OF

A BUILDING AT ONE WITH ITS ENVIRONMENT.

BUT FROM A GEOMETRIC POINT OF VIEW, "FALLING WATER"

ALSO PROVIDES A CONTRAST TO ITS SURROUNDINGS.

THE IRREGULAR SHAPES AND SURFACES OF THE

SURROUNDING ROCKS AND TREES CONTRAST TO THE LINES,

PLANES AND QUADRILATERALS OF THE ARCHITECTURE.

PLANE GEOMETRY EMERGES FROM

THE ROUGH AND IRREGULAR.

IN PARTICULAR, "FALLING WATER" CONSISTS OF

NUMEROUS FOUR SIDED SHAPES.

PERHAPS ITS MOST DISTINCTIVE FEATURE

IS THE BALCONY THAT JUTS OUT OVER THE WATER

SEEMING TO DEFY GRAVITY.

HOW WAS FRANK LOYD WRIGHT ABLE TO ACHIEVE THIS EFFECT?

WHY DID HE CHOOSE RECTANGULAR SHAPES?

YOU'VE SEEN HOW TRIANGULAR FORMS

ARE USED BY ARCHITECTS TO BUILD STURDY BUILDINGS.

IN TERMS OF AREA, YOU'VE ALSO SEEN HOW,

FOR ANY GIVEN SIDE LENGTHS, THE TRIANGLE

WITH A MAXIMUM AREA IS A RIGHT TRIANGLE.

WE CAN NOW TAKE THESE PROPERTIES OF TRIANGLES

AND BRING THEM TO OUR STUDY OF QUADRILATERALS.

HERE'S HOW.

YOU CAN TAKE ANY TRIANGLE AND USE IT TO

CONSTRUCT A QUADRILATERAL.

TAKE A PENCIL AND RULER AND DRAW A TRIANGLE.

DON'T WORRY ABOUT THE ANGLE MEASURES OR SIDE LENGTHS.

USE THIS TRIANGLE AS A TEMPLATE.

TRACE SEVERAL VERSIONS OF THIS SAME TRIANGLE,

CUT THEM OUT TO FORM TWO CONGRUENT TRIANGLES.

IF YOU ALIGN THE TRIANGLES

ALONG ANY PAIR OF CONGRUENT SIDES

YOU'LL FORM A QUADRILATERAL.

A QUADRILATERAL HAS FOUR SIDES, FOUR VERTICES,

AND IS A CLOSED FIGURE.

THESE ARE EXAMPLES OF QUADRILATERALS...

AND THESE ARE NOT.

WE KNOW THAT ANY TWO CONGRUENT TRIANGLES

CAN FORM A QUADRILATERAL.

WE ALSO KNOW THAT FOR ANY GIVEN SET OF SIDES

THE TRIANGLE WITH THE MAXIMUM AREA

IS A RIGHT TRIANGLE.

DOES THIS MEAN THAT TO GET A QUADRILATERAL

WITH A MAXIMUM AREA YOU SHOULD TAKE TWO CONGRUENT

RIGHT TRIANGLES AND JOIN THEM ALONG THE HYPOTENUSE?

DOES A RECTANGLE INHERIT THE RIGHT TRIANGLE'S

MAXIMUM AREA?

LET'S USE THE TI-NSPIRE TO EXPLORE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPHS AND GEOMETRY WINDOW.

YOU WILL BE CREATING A QUADRILATERAL

USING THE POLYGON TOOL.

PRESS MENU AND UNDER SHAPES SELECT POLYGON.

USE THE NAV PAD TO MOVE THE POINTER

TOWARD THE MIDDLE LEFT PART OF THE SCREEN.

PRESS ENTER TO DEFINE THE FIRST VERTEX.

THIS WILL BE THE LOWER LEFT-HAND CORNER

OF THE QUADRILATERAL.

MOVE THE POINTER UP,

THEN PRESS ENTER TO DEFINE THE SECOND VERTEX.

YOU'LL SEE THE FIRST SIDE OF THE QUADRILATERAL APPEAR.

NOW MOVE THE POINTER TO THE RIGHT

TO DEFINE THE THIRD POINT.

PRESS ENTER.

NOTICE THAT THE TOOL WANTS TO CREATE A TRIANGLE,

BUT BY DEFINING THE FOURTH CORNER POINT

YOU'LL CREATE A QUADRILATERAL.

SO MOVE THE POINTER DOWN TO DEFINE THE

LOWER RIGHT-HAND CORNER OF THE QUADRILATERAL.

PRESS ENTER.

FINALLY, MOVE THE POINTER TO THE FIRST POINT

YOU CREATED TO COMPLETE THE CLOSED FIGURE.

PRESS ENTER.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW MEASURE THE LENGTHS OF EACH SIDE.

THIS IS IN ORDER TO ARRANGE THE QUADRILATERAL

TO HAVE SPECIFIC SIDE LENGTHS.

SO PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER TO EACH SIDE TO MEASURE IT.

YOU'LL SEE THE MEASUREMENT APPEAR,

SO PRESS ENTER ONCE TO RECORD THE MEASUREMENT.

THEN MOVE THE POINTER SO THAT IT IS

TO THE SIDE OF THE SEGMENT AND PRESS ENTER AGAIN

TO PLACE THE MEASUREMENT ON-SCREEN.

DO THIS FOR EACH SIDE OF THE QUADRILATERAL.

WHEN YOU ARE DONE, PRESS THE ESCAPE KEY.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW THAT YOU KNOW THE SIDE LENGTHS YOU CAN

MANIPULATE THE VERTICES OF THE QUADRILATERAL

SO THAT THE SIDES ARE A SPECIFIC LENGTH.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE LOWER LEFT-HAND POINT.

YOU'LL SEE THE ON-SCREEN LABEL POINT.

THE POINTER SHOULD LOOK LIKE AN OPEN HAND.

PRESS AND HOLD THE CLICK KEY

UNTIL THE OPEN HAND BECOMES A CLOSED HAND

WHICH INDICATES THAT YOU HAVE SELECTED THE POINT.

MOVE THE POINTER SO THAT THE VERTICAL SIDE

IS AS CLOSE TO 90 DEGREES AS YOU CAN GET IT.

ALSO, CHANGE THE SIDE LENGTH SO THAT IT IS FIVE.

PRESS ESCAPE WHEN YOU ARE DONE.

REPEAT THIS PROCESS WITH THE OTHER THREE POINTS.

YOU WANT TO END UP WITH A RECTANGLE THAT

LOOKS LIKE THIS AND HAS THESE SIDE MEASURES.

DON'T FORGET TO PRESS ESCAPE AFTER YOU HAVE

MOVED EACH OF THE VERTICES TO ITS APPROPRIATE SPOT.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW THAT YOU HAVE SPECIFIED THE SIDE LENGTHS

YOU'LL WANT TO LOCK THESE VALUES

SO THAT AS YOU MANIPULATE THE VERTICES

THE SIDE LENGTHS WON'T CHANGE.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE MEASUREMENT OF THE LEFT-HAND VERTICAL SIDE.

YOU'LL SEE THE ON-SCREEN LABEL "TEXT".

PRESS CONTROL AND THE MENU KEY.

FROM THE DROP DOWN LIST SELECT ATTRIBUTES.

YOU'LL SEE AN ICON THAT LOOKS LIKE AN OPEN LOCK.

USE THE DOWN ARROW TO HIGHLIGHT THIS ICON.

USE THE RIGHT ARROW TO CHANGE THE OPEN LOCK

TO A CLOSED LOCK AND PRESS ENTER.

THE SIDE LENGTH IS NOW LOCKED IN PLACE.

YOU CAN MOVE THE ENDPOINTS OF THE SIDE

BUT THE LENGTH WON'T CHANGE.

REPEAT THIS PROCESS FOR THE OTHER VERTICAL SIDE

AND THE BASE OF THE QUADRILATERAL.

DO NOT CHANGE THE TOP SIDE.

FOR EACH OF THE TWO REMAINING SIDES

PRESS CONTROL AND MENU, SELECT ATTRIBUTES,

AND CHANGE THE OPEN LOCK TO A CLOSED LOCK.

YOU NOW HAVE A QUADRILATERAL

WITH SIDE LENGTHS OF FOUR AND FIVE.

YOU WANT TO CALCULATE THE AREA OF THE QUADRILATERAL.

IN GENERAL, TO FIND THE AREA OF A QUADRILATERAL,

DRAW A DIAGONAL TO SPLIT THE QUADRILATERAL

INTO TWO TRIANGLES THAT SHARE A COMMON SIDE.

THESE TRIANGLES ARE NOT NECESSARILY CONGRUENT

BUT IF YOU ADD THE AREA OF EACH TRIANGLE

YOU WILL FIND THE AREA OF THE QUADRILATERAL.

IF YOU CONSIDER THE COMMON SIDE THE BASE OF EACH

TRIANGLE, THEN THE AREA OF EACH TRIANGLE IS ONE HALF

TIMES THE BASE AND ITS CORRESPONDING HEIGHT.

SO THE GENERAL FORMULA FOR FINDING THE AREA

OF A QUADRILATERAL IS SUMMARIZED BY THIS:

ONE-HALF B TIMES THE QUANTITY H1 PLUS H2.

IF THIS IS A PARALLELOGRAM THEN THE TRIANGLES ARE

CONGRUENT AND H1 AND H2 ARE EQUAL TO EACH OTHER.

THE FORMULA THEN BECOMES BASE TIMES HEIGHT.

THIS IS THE FORMULA WE WILL BE USING.

THE HEIGHT IS DEFINED AS THE LENGTH OF THE SEGMENT

PERPENDICULAR TO THE BASE

THAT INTERSECTS THE OPPOSITE SIDE.

LET'S CONSTRUCT THE BASE.

PRESS MENU AND UNDER "CONSTRUCTION"

SELECT PERPENDICULAR.

THE POINTER SHOULD NOW LOOK LIKE A PENCIL.

MOVE THE POINTER TO THE BASE.

YOU'LL SEE THE SEGMENT THAT DEFINES THE BASE

HIGHLIGHTED AS THE POINTER CHANGES TO A POINTING FINGER.

PRESS ENTER ONCE TO HIGHLIGHT THE SEGMENT.

PRESS ENTER AGAIN TO PLACE A POINT ON THE BASE.

YOU'LL SEE THE PERPENDICULAR LINE AT THIS POINT APPEAR.

MOVE THE POINTER STRAIGHT UP TO REACH THE OPPOSITE SIDE.

WHEN THE POINTER CHANGES TO A POINTING FINGER,

PRESS ENTER TO IDENTIFY A SEGMENT

AND PRESS ENTER AGAIN TO DEFINE

THE ENDPOINT OF THE SEGMENT.

MEASURE THE LENGTH OF THIS SEGMENT.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER TO ONE OF THE ENDPOINTS

OF THE PERPENDICULAR SEGMENT.

PRESS ENTER.

MOVE THE POINTER TO THE OTHER ENDPOINT

OF THE SEGMENT. PRESS ENTER.

YOU'LL SEE THE MEASUREMENT APPEAR.

PRESS ENTER.

MOVE THE POINTER TO THE SIDE OF THE

PERPENDICULAR SEGMENT.

PRESS ENTER AGAIN.

YOU NOW HAVE ALL THE MEASUREMENTS YOU NEED

TO CALCULATE THE AREA OF THE QUADRILATERAL.

BUT ONE MORE MEASUREMENT IS NEEDED

TO SEE THE EFFECT ON THE AREA.

ACTIVATE THE ANGLE MEASUREMENT TOOL.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

NOW MOVE THE POINTER ABOVE THE THREE POINTS ON THE

QUADRILATERAL THAT DEFINE THE LOWER LEFT-HAND VERTEX.

PRESS THE ENTER KEY OVER EACH POINT.

YOU WILL SEE AN ANGLE MEASUREMENT.

PRESS ENTER TO RECORD THE MEASUREMENT.

THEN MOVE THE POINTER TO A CLEAR PART OF THE SCREEN

AND PRESS ENTER ONE MORE TIME.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU WILL BE MANIPULATING THE VERTICES OF THE

QUADRILATERAL TO SEE THE EFFECT ON THE AREA.

FIRST, CREATE A FORMULA FOR CAPTURING

THE AREA MEASUREMENT DATA.

PRESS MENU AND UNDER ACTIONS SELECT TEXT.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN.

PRESS ENTER.

AT THE CURSOR, INPUT THE FORMULA B X H

AND PRESS ENTER.

TO LINK THIS FORMULA TO THE AREA OF THIS QUADRILATERAL

PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.

MOVE THE POINTER SO THAT IT HOVERS OVER THE

FORMULA FOR THE AREA OF THE QUADRILATERAL.

PRESS ENTER.

YOU WILL BE ASKED TO LINK THE FORMULA

TO THE VALUE FOR B.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE LENGTH OF THE BASE.

PRESS ENTER.

THEN MOVE THE POINTER SO THAT IT HOVERS OVER

THE MEASUREMENT OF THE ALTITUDE.

PRESS ENTER AGAIN.

PRESS ESCAPE TO EXIT MEASUREMENT MODE.

NOW MOVE THE POINTER TO THE UPPER RIGHT-HAND

POINT OF THE QUADRILATERAL.

PRESS AND HOLD THE CLICK KEY, WHICH IS THE

ROUND BUTTON AT THE MIDDLE OF THE NAV PAD.

THE POINTER SHOULD CHANGE FROM AN OPEN HAND

TO A CLOSED HAND INDICATING THAT THE POINT IS SELECTED.

USE THE NAV PAD TO MOVE THE POINT AND CHANGE THE

ANGLE MEASURE AND SEE THE EFFECT ON THE AREA.

SINCE YOU DIDN'T LOCK THE MEASUREMENT OF THE SIDE

OPPOSITE THE BASE, ADJUST THE POSITION

OF THIS POINT TO MAINTAIN THE LENGTH.

NOTICE HOW THE AREA CHANGES

AS YOU MANIPULATE THE QUADRILATERAL

EVEN THOUGH THE SIDE LENGTHS ARE CONSTANT.

YOU'LL SEE THAT THE MAXIMUM AREA FOR THE QUADRILATERAL

OF GIVEN SIDE LENGTHS IS INDEED THE RECTANGLE

WHOSE CORNER ANGLES ARE 90 DEGREES.

ALL OTHER QUADRILATERALS

WHICH ARE PARALLELOGRAMS HAVE A SMALLER AREA.

WHEN DESIGNING A HOUSE

YOU WANT THE MAXIMUM AREA POSSIBLE,

GIVEN THE EXPENSE IN BUILDING A HOUSE.

THIS IS WHY MOST HOUSES HAVE RECTANGULAR SPACES.

ANY OTHER KIND OF QUADRILATERAL

WOULD HAVE LESS AREA AND SO WOULD, BY COMPARISON,

BE MORE EXPENSIVE TO BUILD THAN A RECTANGULAR

SHAPED HOUSE OF THE SAME AREA.

LET'S LOOK AT RECTANGLES MORE CLOSELY.

EVERY RECTANGLE IS MADE UP OF TWO CONGRUENT

RIGHT TRIANGLES.

ALL FOUR CORNERS OF THE RECTANGLE ARE RIGHT ANGLES

AND OPPOSITE SIDES ARE CONGRUENT.

FURTHERMORE, THE OPPOSITE SIDES OF A RECTANGLE

ARE PARALLEL. THE REASON FOR THAT

IS THAT THE ALTERNATE INTERIOR ANGLES,

IN OTHER WORDS THE ANGLES AT ADJACENT VERTICES,

ADD UP TO 180 DEGREES.

IN FACT, A RECTANGLE IS A TYPE OF PARALLELOGRAM.

WITH A PARALLELOGRAM OPPOSITE SIDES ARE PARALLEL

BUT THE ANGLES AT EACH VERTEX

ARE NOT NECESSARILY 90 DEGREES.

AS YOU SAW FROM YOUR EXPLORATION,

A PARALLELOGRAM HAS A SMALLER AREA

THAN THE CORRESPONDING RECTANGLE.

ONE OF THE MOST DRAMATIC FEATURES OF "FALLING WATER"

IS THE BALCONY THAT SEEMS TO FLOAT ON AIR.

THIS RECTANGULAR SHAPED AREA

IS MADE OF HEAVY SLABS OF CONCRETE.

HOW IS THIS BALCONY SUPPORTED?

AND HOW CAN WE UNDERSTAND

WHAT'S HAPPENING GEOMETRICALLY?

THE BALCONY IS AN EXAMPLE OF A CANTILEVER.

WITH A CANTILEVER, SUPPORT IS PROVIDED AT ONLY ONE END.

THINK OF A DIVING BOARD AS AN EXAMPLE OF A CANTILEVER.

ONE END OF THE DIVING BOARD IS SUPPORTED

AND THE SUPPORT IS STRONG ENOUGH

TO HOLD THE ADDITIONAL WEIGHT OF A DIVER.

IF YOU'VE EVER HELD AN OBJECT AT ARM'S LENGTH

YOU REALIZE QUICKLY HOW MUCH FORCE IS NEEDED

TO KEEP YOUR ARMS OUTRIGHT.

THE CANTILEVER MUST PROVIDE ENOUGH SUPPORT

FOR THE ENTIRE WEIGHT OF THE BALCONY.

SUPPOSE YOU WANT TO BUILD A BALCONY

BASED ON THIS RECTANGLE.

YOUR INTUITIVE UNDERSTANDING OF GEOMETRY

MAY LEAD YOU TO CONCLUDE THAT

THIS TYPE OF BALCONY TAKES MORE SUPPORT

THAN THIS TYPE OF BALCONY EVEN THOUGH

BOTH BALCONIES HAVE THE SAME AREA.

IF SO, YOU'D BE RIGHT.

BUT WHAT IS HAPPENING GEOMETRICALLY

TO MAKE THIS HAPPEN?

A KEY CONCEPT THAT ARCHITECTS AND BUILDERS

HAVE TO DEAL WITH IS CENTER OF GRAVITY.

THE CENTER OF GRAVITY IS THE POINT IN SPACE

WHERE THE WEIGHT OF AN OBJECT IS BALANCED.

YOU COULD THEORETICALLY BALANCE AN OBJECT

ON ITS CENTER OF GRAVITY.

TO FIND THE CENTER OF GRAVITY OF A QUADRILATERAL,

FIND THE MIDPOINT OF EACH SIDE, THEN CONNECT THE

MIDPOINTS FROM OPPOSITE SIDES WITH A LINE SEGMENT.

THE POINT OF INTERSECTION IS THE CENTER OF GRAVITY.

LET'S EXPLORE CENTER OF GRAVITY ON THE NSPIRE.

CREATE A NEW GEOMETRY WINDOW.

PRESS THE HOME KEY AND SELECT A GRAPHS

AND GEOMETRY WINDOW.

USE THE SEGMENT TOOL TO CREATE A QUADRILATERAL.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

AS YOU'VE DONE BEFORE, MOVE THE POINTER

TO THE MIDDLE PART OF THE SCREEN

AND PRESS ENTER TO DEFINE THE FIRST ENDPOINT.

MOVE THE POINTER TO DEFINE THE SECOND

ENDPOINT OF ONE SIDE OF THE QUADRILATERAL.

PRESS ENTER AND THEN IMMEDIATELY

PRESS ENTER AGAIN TO DEFINE THE STARTING POINT

OF THE NEXT SIDE OF THE QUADRILATERAL.

REPEAT UNTIL YOU HAVE CREATED THE CLOSED FIGURE.

NOW FIND THE MIDPOINT OF EACH SEGMENT.

PRESS MENU AND UNDER CONSTRUCTION

SELECT MIDPOINT.

MOVE THE POINTER ABOVE EACH SIDE OF

THE QUADRILATERAL AND PRESS ENTER EACH TIME.

WHEN YOU DO THAT THE MIDPOINT OF THE SEGMENT

IS CONSTRUCTED.

YOUR SCREEN SHOULD LOOK LIKE THIS.

NOW CONSTRUCT A LINE SEGMENT

CONNECTING OPPOSITE MIDPOINTS.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

MOVE THE POINTER ABOVE ONE OF THE MIDPOINTS

AND PRESS ENTER.

DO NOT CREATE A NEW POINT BUT SIMPLY START YOUR

SEGMENT AT THE MIDPOINT THAT IS ALREADY THERE.

MOVE THE POINTER TO THE OPPOSITE MIDPOINT

AND HOVER OVER THAT POINT.

PRESS ENTER TO CONSTRUCT THE SEGMENT.

REPEAT WITH THE OTHER PAIR OF MIDPOINTS.

PRESS ESCAPE WHEN YOU ARE DONE.

YOU WILL MANIPULATE THE VERTICES

TO CREATE DIFFERENT TYPES OF QUADRILATERALS

TO SEE THE EFFECT ON THE CENTER OF GRAVITY.

START BY MOVING THE VERTICES TO FORM A SQUARE.

DON'T WORRY IF THE SIDE OR ANGLE MEASURES AREN'T EXACT.

WITH A SQUARE, THE CENTER OF GRAVITY

IS IN THE MIDDLE OF THE SQUARE

WHERE THE DIAGONALS OF THE VERTICES INTERSECT.

NOW CONSTRUCT A RECTANGLE BY PUSHING

ONE OF THE SIDES HORIZONTALLY.

NOTICE THAT THE CENTER OF GRAVITY

SHIFTS HORIZONTALLY TOO.

GO BACK TO THE SQUARE SHAPE.

THIS TIME CREATE A RECTANGLE

BY PUSHING ONE OF THE SIDES VERTICALLY.

NOTICE THAT THE CENTER OF GRAVITY CHANGES

IN THE DIRECTION OF THE EXPANSION.

SO WHEN A BALCONY IS ARRANGED THIS WAY,

THE CENTER OF GRAVITY

IS CLOSER TO THE WALL WITH THE CANTILEVER

THAN WHEN A BALCONY IS ARRANGED THIS WAY.

YOU CAN SEE THAT WHEN A BALCONY IS ARRANGED

THIS WAY, THE DISTANCE FROM THE CENTER OF GRAVITY

TO THE CANTILEVER WALL INCREASES.

AS A RESULT MORE FORCE IS REQUIRED

TO SUPPORT THE BALCONY.

THE FORCE ACTING ON THE BALCONY IS KNOWN AS TORQUE.

TORQUE IS BASICALLY THE WEIGHT OF THE BALCONY

TIMES THE DISTANCE FROM THE CENTER OF GRAVITY TO THE WALL.

THE LONGER THE BALCONY THE MORE OF A DOWNWARD TORQUE.

THIS MEANS THAT THE CANTILEVER HAS TO BE

STRONGER TO OFFER MORE SUPPORT.

IN THE CASE OF THE "FALLING WATER" BALCONY,

FRANK LLOYD WRIGHT USED REINFORCED CONCRETE.

METAL BARS KNOWN AS REBAR WERE ENCASED IN THE CONCRETE

TO MAKE THE CANTILEVER SUPPORT THAT MUCH STRONGER.

YET OVER TIME THE BALCONY NEEDED STRUCTURAL REPAIRS

DUE TO THE WEIGHT OF THE BALCONY.

SO QUADRILATERAL FORMS ARE A KEY PART

OF OUR DAY-TO-DAY LIVES

BECAUSE WE LIVE AROUND SQUARES AND RECTANGLES.

THESE SHAPES PROVIDE THE GREATEST AREA

FOR ANY GIVEN SET OF SIDE LENGTHS.

ARCHITECTS TAKE ADVANTAGE OF THESE FORMS TO CREATE

SOME DRAMATIC WORKS THAT GO BEYOND THE DOMESTIC.

IN THE CENTER OF MADRID IS A SET OF CROOKED TOWERS

THAT LOOK LIKE THEY JUST MIGHT COLLAPSE.

THE PUERTA DE EUROPA, OR GATEWAY TO EUROPE,

IS A STYLISH DESIGN THAT IS ALSO A BIT JARRING.

SPAIN'S LOCATION AT ONE END OF EUROPE

AT ONE END OF THE MEDITERRANEAN SEA,

MAKES THE GATEWAY TOWERS SYMBOLIC OF A NEW EUROPE.

AND THE TWO SLANTED TOWERS

ARE MEANT TO SUGGEST A GATEWAY

WITH THE REMINDER OF THE POST AND LINTEL

CONSTRUCTION OF THE ANCIENT WORLD.

OUR UNDERSTANDING OF THE GEOMETRY OF BUILDINGS

TELLS US THAT A BUILDING THAT ISN'T COMPLETELY VERTICAL

IS UNBALANCED AND HEAVIER ON ONE SIDE.

IN THE PREVIOUS SECTION YOU LEARNED ABOUT THE ROLE OF

THE CENTER OF GRAVITY.

MOST TALL BUILDINGS ARE RECTANGULAR IN SHAPE

AND THIS MEANS THAT THEIR CENTER OF GRAVITY IS FOUND

WHERE THE DIAGONALS OF THE RECTANGLE MEET.

BUT WHAT HAPPENS WHEN THE BUILDINGS ARE SLANTED

AS THEY ARE IN MADRID?

GEOMETRICALLY, WHAT'S HAPPENING

IS THAT THE RECTANGULAR SHAPE OF THE BUILDING

IS TRANSFORMED INTO THE SHAPE OF A PARALLELOGRAM.

WITH A PARALLELOGRAM OPPOSITE SIDES ARE PARALLEL

AND ANY PAIR OF ADJACENT ANGLES ARE SUPPLEMENTARY,

MEANING THAT THE SUM OF THE ANGLE MEASURES

IS 180 DEGREES.

WHEN A PARALLELOGRAM STANDS ON ITS BASE

THE CENTER OF GRAVITY LEANS IN THE DIRECTION

OF THE PARALLELOGRAM'S SLANT.

A FREESTANDING OBJECT WILL TIP OVER

WHEN THE ANGLE FORMED BY THE CENTER OF GRAVITY

AND THE BASE OF THE OBJECT IS GREATER THAN 90 DEGREES.

AT THAT POINT THE OBJECT IS OUT OF BALANCE

AND WILL FALL OVER.

IF AN OBJECT IS ANCHORED TO THE GROUND,

THE ANGLE FORMED BY THE CENTER OF GRAVITY

CAN BE GREATER THAN 90 DEGREES

WITHOUT THE OBJECT TIPPING OVER.

BUT THIS IS HARDLY EVER A PREFERRED WAY OF BUILDING.

TOO MUCH IMBALANCE WOULD MAKE FOR AN

UNSTEADY BUILDING.

THE PUERTA DE EUROPA TOWERS ARE TILTED

BUT IN SUCH A WAY THAT ITS CENTER OF GRAVITY

IS NOT GREATER THAN 90 DEGREES.

GIVEN THE HEIGHT AND WIDTH OF THE TOWERS,

WHAT IS THE MAXIMUM ANGLE THAT THE BUILDINGS CAN TILT?

AND WHAT IS THE ACTUAL ANGLE OF TILT FOR THE BUILDINGS?

LET'S USE THE TI-NSPIRE TO EXPLORE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPHS AND GEOMETRY WINDOW.

YOU WANT TO CREATE A PARALLELOGRAM.

START BY CREATING A LINE.

PRESS THE MENU BUTTON AND UNDER "POINTS AND LINES"

SELECT LINE.

START BY CONSTRUCTING THE BASELINE.

MOVE THE POINTER TO THE LOWER LEFT-HAND

PART OF THE SCREEN, LEAVING ENOUGH WHITE SPACE

TO THE LEFT AND BELOW WHERE THE LINE WILL BE.

PRESS ENTER.

YOU'LL SEE THAT A POINT IS ADDED

AND THE OUTLINE OF A LINE APPEARS.

USE THE RIGHT ARROW TO MOVE THE POINTER TO THE RIGHT.

MOVE IT ALL THE WAY TO THE OTHER SIDE OF THE SCREEN.

PRESS ENTER TO DEFINE THE LINE.

YOU WANT TO CREATE A LINE PARALLEL TO THIS LINE.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PARALLEL.

MOVE THE POINTER ABOVE THE LINE YOU JUST CREATED

AND PRESS ENTER.

YOU'LL SEE A PARALLEL LINE APPEAR.

NEXT USE THE UP ARROW TO MOVE THE POINTER

TO THE TOP OF THE SCREEN,

LEAVING ENOUGH ROOM ABOVE THE LINE.

PRESS ENTER TO DEFINE THE PARALLEL LINE.

NOW CREATE A PAIR OF PARALLEL LINES THAT

INTERSECT THE PARALLEL LINES YOU'VE JUST CREATED.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT LINE.

MOVE THE POINTER ABOVE ONE OF THE POINTS YOU CREATED.

PRESS ENTER.

MOVE THE POINTER DOWN AND TO THE RIGHT

TO CREATE A SLANTED INTERSECTING LINE.

WHEN THE POINTER IS ABOVE THE SECOND LINE,

PRESS ENTER.

MAKE SURE THAT AN INTERSECTION POINT

IS CREATED.

NOW CONSTRUCT THE LINE PARALLEL TO THIS LINE.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PARALLEL.

MOVE THE POINTER ABOVE THE LINE YOU JUST CREATED.

PRESS ENTER.

MOVE THE POINTER SO THAT IT IS

TO THE RIGHT OF THE FIRST LINE,

BUT ALSO MAKE SURE THE POINTER IS AT A PLACE WHERE

THE NEW LINE INTERSECTS ONE OF THE OTHER LINES.

PRESS ENTER TO CREATE AN INTERSECTION POINT.

YOU NOW SHOULD HAVE THREE OF THE VERTICES

OF THE PARALLELOGRAM.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER ABOVE THE PARALLEL LINE

YOU'VE JUST CREATED.

PRESS ENTER.

MOVE THE POINTER ABOVE THE REMAINING LINE

THAT NEEDS AN INTERSECTION POINT

AND PRESS ENTER ONCE MORE.

YOU NOW HAVE A PARALLELOGRAM

YOU CAN USE TO MODEL THE SHAPE OF

ONE OF THE PUERTA DE EUROPA TOWERS.

IF YOU MODIFY ANY OF THE VERTICES

THE SHAPE OF THE TOWER WILL CHANGE

BUT THE SIDES WILL REMAIN PARALLEL.

YOU NOW WANT TO FIND THE CENTER OF GRAVITY

OF THIS PARALLELOGRAM.

RECALL THAT THIS INVOLVES FINDING

THE MIDPOINT OF EACH SIDE.

PRESS MENU AND UNDER CONSTRUCTION

SELECT MIDPOINT.

MOVE THE POINTER ABOVE ONE OF THE VERTICES

OF THE PARALLELOGRAM.

PRESS ENTER.

THEN MOVE THE POINTER TO THE ADJACENT VERTEX

AND PRESS ENTER AGAIN.

NOTICE THAT THE MIDPOINT IS NOW

ON THE PARALLELOGRAM SIDE.

REPEAT THESE STEPS FOR THE REMAINING

THREE SIDES OF THE PARALLELOGRAM.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NEXT CONSTRUCT SEGMENTS

CONNECTING OPPOSITE MIDPOINTS.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

MOVE THE POINTER ABOVE ONE OF THE MIDPOINTS

AND PRESS ENTER.

MOVE THE POINTER TO THE OPPOSITE MIDPOINT

AND PRESS ENTER AGAIN.

REPEAT THIS FOR THE OTHER PAIR OF MIDPOINTS.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NEXT, PLACE A POINT WHERE THE TWO SEGMENTS INTERSECT.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER ABOVE ONE OF THE SEGMENTS

AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE OTHER SEGMENT

AND PRESS ENTER AGAIN.

NOW THAT YOU''VE FOUND THE CENTER OF GRAVITY

OF THE BUILDING, FIND THE ANGLE MEASURE

FORMED BY THE CENTER OF GRAVITY,

THE LOWER LEFT-HAND CORNER VERTEX,

AND THE BASE OF THE PARALLELOGRAM.

FIRST, CONSTRUCT A LINE SEGMENT

CONNECTING THE CENTER OF GRAVITY TO THE VERTEX.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

DRAW A SEGMENT CONNECTING THE POINTS AS SHOWN.

NOW MEASURE THE ANGLE BETWEEN THIS SEGMENT

AND THE BASE OF THE PARALLELOGRAM.

PRESS MENU AND UNDER MEASUREMENT

SELECT ANGLE.

MOVE THE POINTER ABOVE THE CENTER OF GRAVITY

AND PRESS ENTER.

MOVE THE POINTER TO THE CORNER VERTEX

AND PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER TO THE POINT ON THE BASE

AND PRESS ENTER.

YOU WILL NOW SEE AN ANGLE MEASURE.

AS YOU CHANGE THE ORIENTATION OF THE

PARALLELOGRAM, THE ANGLE MEASURE CHANGES.

THE LIMIT ON HOW FAR YOU CAN ORIENT THE TOWER

IS 90 DEGREES.

TILTING THE TOWER SO THAT THE ANGLE MEASURE

IS GREATER THAN 90 DEGREES

CREATES AN UNMANAGEABLE IMBALANCE.

TO FIND THE AMOUNT THE BUILDING HAS TILTED,

MEASURE THE EXTERIOR ANGLE THAT THE BUILDING

MAKES WITH THE GROUND.

SO ONCE AGAIN PRESS MENU

AND UNDER MEASUREMENT SELECT ANGLE.

MEASURE THE EXTERIOR ANGLE AS SHOWN.

A STRAIGHT TOWER HAS A 90 DEGREE ANGLE

BUT A TILTED ONE IS GREATER THAN 90 DEGREES

BASED ON HOW THIS TOWER IS BEING TILTED.

SIMPLY CALCULATE THE AMOUNT BEYOND 90 DEGREES

TO DETERMINE THE TILT OF THE TOWER.

BUT NOTICE THAT IF YOU INCREASE THE HEIGHT OF

THE TOWER THE AMOUNT OF TILT CAN ALSO INCREASE.

SO A REALLY TALL TOWER CAN TILT EVEN MORE

AND CREATE A DRAMATIC EFFECT

WHILE STILL MAINTAINING A MANAGEABLE IMBALANCE.

IN THE CASE OF THE PUERTA DE EUROPA TOWERS

YOU DON'T EVEN NEED THE MEASUREMENTS.

LOOK AT THE DESIGN OF THE TOWER.

NOTICE THAT THE HEIGHT OF THE TOWER

IS FOUR TIMES LONGER THAN THE WIDTH OF THE TOWER.

WE CAN USE THIS INFORMATION

TO MODEL THE TOWER AND FIND THE ANGLE OF TILT.

YOU FIRST NEED TO MEASURE THE LENGTHS OF THE SIDES

OF THE PARALLELOGRAM YOU CONSTRUCTED.

PRESS MENU AND UNDER MEASUREMENTS

SELECT LENGTH.

MOVE THE POINTER TO ONE OF THE VERTICES

THAT DEFINE THE BASE OF THE PARALLELOGRAM.

PRESS ENTER.

NEXT MOVE THE POINTER TO THE OTHER VERTEX

THAT DEFINES THE BASE OF THE PARALLELOGRAM

AND PRESS ENTER AGAIN.

YOU'LL SEE THE MEASUREMENT OF THE LENGTH.

PRESS ENTER TO RECORD THIS MEASUREMENT.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN

AND PRESS ENTER AGAIN

TO PLACE THE MEASUREMENT ON-SCREEN.

REPEAT THIS WITH THE VERTICAL SIDE

OF THE PARALLELOGRAM.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW MANIPULATE THE UPPER LEFT-HAND VERTEX

OF THE PARALLELOGRAM.

TO SELECT IT, HAVE THE POINTER HOVER OVER

THE VERTEX AND CLICK AND HOLD THE CLICK KEY

SO THAT THE POINTER CHANGES FROM

AN OPEN HAND TO A CLOSED HAND.

THEN USE THE NAV PAD TO MOVE THE VERTEX.

USE THE UP ARROW TO CHANGE THE HEIGHT OF THE TOWER

SO THAT IT IS ROUGHLY FOUR TIMES THE LENGTH OF THE BASE.

USE THE LEFT ARROW TO CHANGE THE ANGLE OF TILT

OF THE TOWER.

STOP WHEN THE ANGLE THE CENTER OF GRAVITY

MAKES WITH THE BASE IS 90 DEGREES.

CHECK THE AMOUNT OF THE BUILDING'S TILT

BY INSPECTING THE OTHER ANGLE.

YOU'LL FIND THAT THE TOWER IS TILTED

AT A 15 DEGREE ANGLE WHICH IN FACT IS WHAT THE

ANGLE MEASURE IS FOR THE ACTUAL TOWERS IN MADRID.

NOTICE THAT THE MAIN VERTICAL LINE IN EACH TOWER

IS ACTUALLY ONE OF THE DIAGONALS

OF THE PARALLELOGRAM FACE.

THIS LINE ALSO CROSSES THE CENTER OF GRAVITY,

SO THE ANGLE FORMED BY THE CENTER OF GRAVITY

AND THE BASE OF THE PARALLELOGRAM

IS EXACTLY 90 DEGREES.

THIS CONFIRMS THAT THE TILT OF THE BUILDING

IS ALSO 15 DEGREES.

THE SPACE BETWEEN THE TWO TOWERS

FORMS A DIFFERENT TYPE OF QUADRILATERAL.

THE TWO HORIZONTAL SIDES ARE PARALLEL

SINCE THEY ARE ON THE SAME LINES THAT

DEFINE THE TOP AND BASE OF THE PARALLELOGRAMS.

HOWEVER, THE SLANTED SIDES ARE NOT PARALLEL.

AS A RESULT, THE SHAPE BETWEEN THE TWO TOWERS

IS A TRAPEZOID.

YOU CAN ALSO SEE TRAPEZOIDAL SHAPES

THROUGHOUT THE DESIGN OF THE BUILDING.

THE PUERTA DE EUROPA TOWERS EXTEND THE USE OF

FAMILIAR RECTANGULAR FORMS IN TALL BUILDINGS

TO ENCOMPASS DIFFERENT QUADRILATERAL FORMS.

WHAT ITS PARALLELOGRAM SHAPE LOSES IN AREA

COMPARED TO A RECTANGULAR VERSION OF THE BUILDING

IT MAKES UP IN A CLEVER DESIGN

THAT IS BOTH MODERN AND ANCIENT AT THE SAME TIME.