Geometry Applications: Paralleograms and Trapezoids

Geometry Applications: Paralleograms and Trapezoids

[Music]

[Music]

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.