Title: Geometry Applications: Composite Figures

Title: Geometry Applications: Composite Figures

[Music]

[Music]

THE PETRONAS TOWERS IN KUALA LUMPUR

ARE THE PRIDE OF MALAYSIA.

THESE GRAND STRUCTURES

ARE THE WORLD'S TALLEST TWIN TOWERS.

THESE 88-FLOOR TOWERS TOOK SEVEN YEARS TO BUILD

AND ARE A TESTAMENT TO MALAYSIA'S ECONOMIC GROWTH

AS WELL AS THE GROWING INFLUENCE

OF THE ASIAN ECONOMY IN THE GLOBAL MARKETPLACE.

THE TWO TOWERS ARE CONNECTED BY

A TWO-LEVEL BRIDGE MIDWAY UP EACH TOWER,

MAKING IT THE HIGHEST TWO LEVEL BRIDGE IN THE WORLD.

PROVIDING A PANORAMIC VIEW OF KUALA LUMPUR,

THE SKYBRIDGE IS ALSO AN EMERGENCY EVACUATION

FROM ONE TOWER TO ANOTHER...

AN ISSUE OF CONCERN IN THE POST-911 WORLD.

MALAYSIA IS A PREDOMINANTLY

MUSLIM COUNTRY AND THIS IS REFLECTED

IN THE ARCHITECTURE OF THE PETRONAS TOWERS.

WHEN VIEWED FROM ABOVE, THE TOWERS REVEAL

A FAMILIAR PATTERN FOUND IN ISLAMIC ART.

NOTE THE OVERLAPPING SQUARES, WHICH IS A COMMON

DESIGN FOUND THROUGHOUT THE ISLAMIC WORLD.

THE CIRCULAR ARCS WERE ADDED TO THE DESIGN

OF THE PETRONAS TOWERS AND WE'LL ANALYZE THESE SHAPES

IN MORE DETAIL LATER.

BUT FOR NOW LET'S FOCUS ON THE OVERLAPPING SQUARES.

AS YOU SAW IN THE PREVIOUS SECTION,

ISLAMIC ARTISANS USED A COMPASS AND STRAIGHTEDGE

TO CREATE ELABORATE PATTERNS THAT WERE PRECISE

BUT COULD ALSO BE REPLICATED QUITE EASILY.

LET'S USE THE TI-NSPIRE TO CONSTRUCT AND EXPLORE THIS

STAR PATTERN MADE UP OF THE TWO OVERLAPPING SQUARES.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW AND MAKE SURE

THAT THE VIEW IS THE PLANE GEOMETRY VIEW.

CREATE A HORIZONTAL LINE SEGMENT.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

NOW CONSTRUCT A LINE

PERPENDICULAR TO THIS LINE SEGMENT.

PRESS MENU AND UNDER "CONSTRUCTION"

SELECT PERPENDICULAR.

MOVE THE POINTER ABOVE THE HORIZONTAL LINE

AND PRESS ENTER.

CONSTRUCT A CIRCLE WHOSE CENTER IS AT THE POINT

WHERE THE TWO LINES INTERSECT.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

AFTER CREATING A CIRCLE, CONSTRUCT TWO MORE CIRCLES

WHOSE CENTERS ARE AT THE POINTS WHERE THE FIRST

CIRCLE AND THE HORIZONTAL LINE SEGMENT INTERSECT.

THESE CIRCLES SHOULD HAVE THE SAME RADIUS AS THE

FIRST CIRCLE AND INTERSECT THAT CIRCLE'S CENTER.

REPEAT THIS PROCESS FOR THE TWO POINTS

WHERE THE VERTICAL LINE INTERSECTS THE FIRST CIRCLE.

IN EACH CASE THE NEW CIRCLE WILL BE CENTERED

AT THE INTERSECTION POINT AND ALSO TOUCH THE CENTER

OF THE FIRST CIRCLE.

YOUR SCREEN SHOULD LOOK LIKE THIS.

MAKE A NOTE OF THESE FOUR HIGHLIGHTED

INTERSECTION POINTS.

USE THE SEGMENT TOOL TO CREATE TWO LINE SEGMENTS

THAT INTERSECT AT THE CENTER OF THE FIRST CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

CONNECT THE PAIRS OF POINTS AS SHOWN.

YOU NOW HAVE ALL THE POINTS NECESSARY

TO CONSTRUCT THE OVERLAPPING SQUARES.

LET'S START WITH THE SQUARE

THAT HAS THE HORIZONTAL BASE.

NOTE THESE FOUR HIGHLIGHTED POINTS.

CONTINUING WITH THE SEGMENT TOOL,

CREATE THE SQUARE DEFINED BY THESE FOUR POINTS.

NEXT, USING THESE FOUR HIGHLIGHTED POINTS

CONSTRUCT THE OTHER SQUARE.

THIS SHAPE, AND VARIATIONS OF IT,

ARE FOUND IN MANY ARABESQUES.

THE OUTLINE OF THIS SHAPE IS KNOWN AS A

COMPOSITE FIGURE.

A COMPOSITE FIGURE IS A COMPLEX SHAPE

MADE UP OF SIMPLER GEOMETRIC SHAPES.

IN THE CASE OF THE ISLAMIC STAR THE PRIMARY SHAPES

INCLUDE THE SQUARE WITH THE HORIZONTAL BASE

AND FOUR ISOSCELES TRIANGLES.

THE TOTAL AREA OF THIS COMPOSITE FIGURE

IS THE SUM OF THE AREA OF THE LARGE SQUARE

AND THE COMBINED AREAS OF THE ISOSCELES TRIANGLES.

BUT HOW CAN WE CALCULATE THE AREA OF THE TRIANGLES?

FOR THAT, LET'S LOOK AT A VARIATION

OF THIS COMPOSITE FIGURE.

RATHER THAN LOOKING AT IT AS A COMBINATION

OF ONE SQUARE AND FOUR TRIANGLES,

NOTE THAT THERE IS ALSO A REGULAR HEXAGON.

LOOKED AT THIS WAY,

THE COMPOSITE FIGURE IS ONE HEXAGON

AND SIX CONGRUENT ISOSCELES TRIANGLES.

LET'S LOOK AT THIS TRIANGLE IN MORE DETAIL.

RECALL THAT FOR AN ISOSCELES TRIANGLE,

TWO SIDES AND TWO ANGLES ARE CONGRUENT.

BUT THIS ISOSCELES TRIANGLE IS A CORNER OF A SQUARE

SO THIS IS A SPECIAL TYPE OF ISOSCELES TRIANGLE...

NAMELY AN ISOSCELES RIGHT TRIANGLE.

THIS IS ALSO KNOWN AS THE 45-45-90 DEGREE

RIGHT TRIANGLE.

THESE RIGHT TRIANGLES HAVE SIDE LENGTHS THAT

FORM THE RATIO ONE, ONE, SQUARE ROOT OF TWO.

SUPPOSE THAT THE SIDE LENGTH OF THIS TRIANGLE IS S.

THEN THE LENGTH OF A HYPOTENUSE

IS SQUARE ROOT OF 2S.

THE AREA OF THIS ISOSCELES TRIANGLE IS ONE HALF BASE

TIMES HEIGHT, OR 1/2 S SQUARED.

GOING BACK TO THE COMPOSITE FIGURE, WE CAN SEE

THAT THE LARGE SQUARES HAVE THESE SIDE LENGTHS.

THE TOTAL AREA OF THE COMPOSITE FIGURE

IS THE SUM OF THE SQUARE AND FOUR TIMES THE AREA

OF ONE OF THE ISOSCELES TRIANGLES.

THE SOLUTION IS SHOWN HERE.

ANOTHER WAY TO CALCULATE THE AREA

IS TO USE THE PROPERTIES OF REGULAR POLYGONS.

FOR ANY REGULAR POLYGON THINK OF A CIRCLE

IN WHICH THE POLYGON IS INSCRIBED.

THE INTERIOR ANGLES ARE CONGRUENT AND THE MEASURE

OF EACH ANGLE IS FOUND USING THIS FORMULA

WHERE N IS THE NUMBER OF SIDES OF THE POLYGON.

THERE ARE N ISOSCELES TRIANGLES.

THE BASE OF EACH TRIANGLE IS ONE OF THE EDGES

OF THE POLYGON OF LENGTH S.

THE ALTITUDE OF EACH TRIANGLE, ALSO REFERRED TO

AS THE APATHEM, IS PERPENDICULAR TO SIDE S.

SO THE AREA OF EACH TRIANGLE IS ONE-HALF AS.

SINCE A IS ONE OF THE LEGS OF A RIGHT TRIANGLE,

THEN YOU CAN SOLVE FOR A IN TERMS OF N AND S.

USE THE PYTHAGOREAN THEOREM AND SOLVE FOR A.

WE GET A EQUALS THE SQUARE ROOT OF THE QUANTITY

R SQUARED MINUS S OVER TWO SQUARED

WHERE R IS THE RADIUS OF THE CIRCLE

AND S IS THE LENGTH OF THE POLYGON SIDE.

USING THIS EXPRESSION FOR A IN THE AREA FORMULA

WE GET THIS EQUATION AND THIS EQUATION CAN BE USED

TO FIND THE AREA OF ANY REGULAR POLYGON.

THIS FORMULA COULD BE USED TO FIND THE AREA

OF THE REGULAR HEXAGON SHOWN EARLIER.

THE DESIGN OF THE PETRONAS TOWERS

USES A VARIATION OF THE ISLAMIC SHAPE.

NOTICE THE ADDITIONAL CIRCULAR SECTIONS.

THESE WERE ADDED TO INCREASE THE AMOUNT OF

OFFICE SPACE AVAILABLE, SO THIS FORMS A NEW

COMPOSITE FIGURE SHOWN HERE.

THE QUESTION BECOMES, HOW MUCH MORE AREA HAS BEEN

ADDED TO THE AREA OF THE ORIGINAL COMPOSITE FIGURE?

LET'S ANALYZE THE FIGURES MORE CLOSELY.

A CLOSE UP SHOWS THAT THE CIRCULAR REGION

IS PART OF A CIRCLE WHOSE CENTER IS AT

THE CORNER OF TWO ADJACENT TRIANGLES.

THE RADIUS EXTENDS HALF THE LENGTH OF THE TRIANGLE SIDE.

THIS MEANS THAT THE CIRCLE HAS A RADIUS OF S OVER 2.

THE AREA OF THE FULL CIRCLE IS PI S SQUARED OVER 4.

BUT ONLY A PORTION OF THE CIRCULAR REGION IS USED.

WHAT PORTION? WE KNOW THAT THE INTERIOR ANGLE

OF THE ISOSCELES TRIANGLE IS 45 DEGREES.

THIS MEANS THAT THE CIRCLE

SWEEPS OUT A REGION OF 135 DEGREES.

SINCE A FULL CIRCLE SWEEPS OUT 360 DEGREES,

THE FRACTIONAL AMOUNT THAT THIS PORTION OF THE CIRCLE

SWEEPS OUT IS 135 OVER 360, OR THREE-EIGHTHS.

SO THE AREA OF ONE CIRCULAR SECTION IS THREE-EIGHTHS

OF PI S SQUARED OVER 4, OR THREE PI S SQUARED OVER 32.

AND NOW WE CAN CALCULATE THE TOTAL AREA OF THE

COMPOSITE FIGURE THAT MAKES UP ONE FLOOR IN EACH TOWER.

TAKE THE TOTAL AREA YOU FOUND FOR THE

OVERLAPPING SQUARE SECTION AND ADD IT

TO THE COMBINED AREAS OF THE CIRCULAR SECTIONS.

THE TOTAL AREA IS THIS EXPRESSION.

WHAT PERCENT INCREASE DOES THE ADDITION

OF THE CIRCULAR SECTIONS REPRESENT?

FOR THAT, LET'S USE THE PERCENT INCREASE FORMULA.

DIVIDE THE TOTAL AREA OF THE CIRCULAR SECTIONS

BY THE TOTAL AREA OF THE OVERLAPPING SQUARES.

YOU CAN CREATE A CALCULATOR WINDOW IN THE NSPIRE

TO CALCULATE THE PERCENT INCREASE.

IT TURNS OUT TO BE A ROUGHLY 17% INCREASE IN AREA.

THIS IS A SIZABLE INCREASE IN THE SQUARE FOOTAGE OF

THE AREA, YET THE ADDITION OF THE CIRCULAR SECTIONS

DOES NOT INTERFERE WITH THE OVERALL DESIGN OF THE TOWER.

AND FROM A DISTANCE THESE CIRCULAR SECTIONS

ADD A SMOOTHNESS TO THE LOOK OF THE TOWERS.

COMPOSITE FIGURES PROVIDE A WAY TO CREATE

A VARIETY OF NEW SHAPES WHILE STILL MAINTAINING

THE PROPERTIES OF THE ORIGINAL SHAPES.

AND THE PETRONAS TOWERS PROVIDE A

SOPHISTICATED USE OF SUCH COMPOSITE FIGURES.