Geometry Applications: Angles

Geometry Applications: Angles

[Music]

[Music]

HIMEJI CASTLE HOLDS A SPECIAL PLACE

IN JAPAN'S HISTORY.

BUILT IN THE MIDDLE AGES, IT HAS SURVIVED

CIVIL WARS AND WORLD WARS.

NOT BAD FOR A WOODEN STRUCTURE.

BUT IT IS THE TIMELESS GEOMETRY OF THIS CASTLE

THAT MAKES IT MOST IMPRESSIVE.

IT IS A CASE STUDY IN THE CREATIVE USE OF ANGLES.

HIMEJI CASTLE SERVED AS A FORTRESS

WHERE SAMURAI ARMIES COULD DEFEND THEMSELVES

FROM AN ATTACKING ARMY.

SO HIMEJI CASTLE WAS BUILT TO MAKE IT DIFFICULT

FOR AN ARMY TO INVADE AND EASIER FOR THOSE

INSIDE THE CASTLE TO DEFEND THEMSELVES.

LET'S LOOK AT HOW ANGLES PLAY A KEY ROLE

IN HIMEJI CASTLE'S DEFENSES.

LET'S FIRST START WITH ITS ALTITUDE.

LIKE MANY CASTLES AND FORTRESSES, HIMEJI IS A

TALL STRUCTURE AND MADE ALL THE MORE TALLER

BECAUSE IT IS SITUATED ON A HILL.

THE TOTAL HEIGHT OF HIMEJI CASTLE IS 92 METERS

OR ABOUT THE LENGTH OF A FOOTBALL FLD.

THE ADVANTAGE OF A TALL STRUCTURE IS THAT

IT INCREASES THE LINE OF SIGHT TO THE HORIZON.

THIS MAKES IT MUCH EASIER TO SEE AN INVADING ARMY

FROM FARTHER AWAY,

ALLOWING THOSE IN THE CASTLE TO PREPARE DEFENSES.

THIS IS HOW LINE OF SIGHT WORKS.

IMAGINE A STRUCTURE SITTING ON THE EARTH'S SURFACE

WHICH IS DRAWN AS A CIRCLE.

DRAW A LINE FROM THE TOP OF THE STRUCTURE

TO A POINT ON THE CIRCLE.

A LINE THAT INTERSECTS A CIRCLE AT ONE POINT

IS TANGENT TO THE CIRCLE.

NOW, IMAGINE A TALLER STRUCTURE.

WE DRAW ANOTHER LINE FROM THE TOP OF THIS STRUCTURE

TO A POINT TANGENT TO THE CIRCLE.

COMPARING THE FIRST LINE OF SITE DIAGRAM TO THE SECOND

YOU'LL SEE THAT AS THE STRUCTURE'S HEIGHT INCREASES

THE MORE OF THE EARTH'S SURFACE IT CAN SEE.

THE LINE OF SIGHT IMPROVES FOR TALLER BUILDINGS.

LET'S EXPLORE THE ANGLES GENERATED FROM LINE OF SIGHT

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.

CREATE A CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN AND

PRESS ENTER TO DEFINE THE CENTER OF THE CIRCLE.

NEXT, MOVE THE POINTER AWAY FROM THE CENTER

TO DEFINE THE SIZE OF THE RADIUS.

PRESS ENTER AGAIN.

CONSTRUCT THE TANGENT TO THE CIRCLE.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT TANGENT.

MOVE THE POINTER TO THE CIRCLE NEAR THE POINT

THAT WOULD CORRESPOND TO 1 O'CLOCK

IF THE CIRCLE WERE A CLOCK FACE.

PRESS ENTER.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

IF YOU NEED TO MOVE THE POINT, SIMPLY HIGHLIGHT IT

AND USE THE NAV PAD TO MOVE THE POINT ALONG THE CIRCLE.

YOU NOW HAVE A LINE TANGENT TO THE CIRCLE.

YOU'LL NOW CONSTRUCT A LINE SEGMENT FROM THE

CENTER OF THE CIRCLE TO A POINT ON THE TANGENT LINE.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE.

THE POINTER SHOULD CHANGE TO A HAND.

PRESS ENTER.

MOVE THE POINTER STRAIGHT UP

UNTIL IT INTERSECTS THE TANGENT LINE.

YOU SHOULD SEE A TEXT MESSAGE NEXT TO THE POINTER

SAYING "POINT ON".

PRESS ENTER.

REPEAT THIS TO CREATE A SEGMENT

FROM THE CENTER OF THE CIRCLE TO THE POINT

WHERE THE TANGENT LINE INTERSECTS THE CIRCLE.

YOU SHOULD NOW HAVE A TRIANGLE

WITH VERTICES AT THE CENTER OF THE CIRCLE,

ON THE CIRCLE ITSELF, AND ON THE TANGENT LINE.

THIS IS A MODEL OF THE LINE OF SIGHT.

THE PORTION OF THE SEGMENT ABOVE THE CIRCLE

REPRESENTS HIMEJI CASTLE.

THE LINE OF SIGHT IS ONE OF THE SIDES OF THE TRIANGLE.

LET'S NOW EXPLORE THE ANGLE MEASURES INVOLVED.

USE THE ANGLE MEASURE TOOL.

PRESS MENU, AND UNDER MEASUREMENT SELECT ANGLE.

WITH THE ANGLE MEASURE TOOL YOU NEED TO DEFINE

THE THREE POINTS THAT MAKE UP THE TRIANGLE.

YOU WILL BE USING THE VERTICES OF THE TRIANGLE.

TAKE TURNS MARKING THE THREE POINTS

THAT MAKE UP EACH OF THE ANGLES

OF THE TRIANGLE YOU HAVE CONSTRUCTED.

PRESS ENTER EACH TIME YOU HAVE HIGHLIGHTED

ONE OF THE POINTS.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

WHEN YOU ARE DONE PRESS THE ESCAPE KEY.

FIRST NOTICE THAT THE ANGLE FORMED BY

THE TANGENT AND THE RADIUS IS 90 DEGREES.

EVEN IF YOU MANIPULATE THE VERTEX

THE ANGLE REMAINS 90 DEGREES.

AND IN FACT THIS IS A PROPERTY OF ALL

TANGENTS TO CIRCLES: THEY ARE PERPENDICULAR

TO THE RADIUS AT THE POINT OF INTERSECTION.

THIS MEANS THAT THE LINE OF SIGHT ANGLE IS AN

ACUTE ANGLE OR AN ANGLE LESS THAN 90 DEGREES.

LET'S CHANGE THE HEIGHT OF A TOWER

TO SEE HOW THE LINE OF SIGHT CHANGES.

MOVE THE POINTER TO THE POINT ON THE TANGENT LINE.

SELECT THE POINT BY PRESSING AND HOLDING THE CLICK KEY.

MOVE THE POINT UPWARD.

NOTICE HOW THE HEIGHT INCREASES.

BECAUSE THE POINT IS ON THE TANGENT LINE,

THE POINT WILL MOVE UP AND TO THE LEFT.

GO TO THE POINT ON THE CIRCLE AND MOVE IT CLOCKWISE

UNTIL THE LINE REPRESENTING HIMEJI CASTLE IS VERTICAL.

THIS MOVEMENT OF THE POINT OF THE CIRCLE

REPRESENTS THE INCREASE IN THE LINE OF SIGHT.

NOTICE ALSO THAT AS THE HEIGHT OF THE STRUCTURE

INCREASES, THE ACUTE ANGLE DECREASES IN SIZE.

HIMEJI CASTLE RISES HIGH IN THE AIR SO THAT IT HAS

A LONG LINE OF SIGHT TO THE HORIZON SO THAT

IT CAN IDENTIFY ATTACKERS FROM A LONG DISTANCE.

THE LINE OF SIGHT FORMS A LINE TANGENT TO THE

ARC OF THE EARTH'S CIRCULAR SHAPE.

WHAT IS THE LENGTH OF THIS LINE OF SIGHT

AND HOW CAN WE USE IT?

BECAUSE THE TANGENT AND THE RADIUS FORM A RIGHT ANGLE

THE TRIANGLE THAT YOU CREATED EARLIER

IS A RIGHT TRIANGLE.

SO THE LINE OF SIGHT DISTANCE

IS ONE OF THE LEGS OF A RIGHT TRIANGLE.

ALSO, THE RADIUS OF THE CIRCLE IS ACTUALLY

THE RADIUS OF THE EARTH.

WE CAN CREATE AN EQUATION TO SOLVE.

LET'S USE VARIABLE H FROM THE HEIGHT OF THE CASTLE.

R FOR THE RADIUS OF THE EARTH

AND L AS THE LINE OF SIGHT DISTANCE.

USING THE PYTHAGOREAN THEOREM

WE GET THE EQUATION L SQUARED PLUS R SQUARED

EQUALS THE QUANTITY H + R SQUARED.

SOLVING FOR L WE GET THE EXPRESSION

THE SQUARE ROOT OF THE QUANTITY H SQUARED +2 HR.

WE KNOW THAT H, THE HEIGHT OF

THE HIMEJI CASTLE IS 92 METERS.

THE RADIUS OF THE EARTH IS ROUGHLY

6.4X10 TO THE 6TH METERS.

PLUGGING THESE VALUES INTO THE FORMULA FOR FINDING L

WE'D GET THAT THE LINE OF SIGHT DISTANCE

IS ABOUT 34,316 METERS, OR ABOUT 21 MILES.

GIVEN THAT AN INVADING ARMY TRAVELS AT A FAIRLY

SLOW SPEED, THE LOOKOUTS ON HIMEJI CASTLE

WITH THEIR COMMANDING VIEW OF THE COUNTRYSIDE

COULD PREPARE FOR IMMINENT INVASION WITH HOURS TO SPARE.

AND WHAT WOULD BE IN STORE FOR THE INVADERS

WOULD BE MORE HOSTILE GEOMETRY.

THE GROUNDS SURROUNDING THE CASTLE ARE VAST

AND THERE ARE PLENTY OF OBSTACLES

ON THE WAY TO THE MAIN TOWER.

FIRST, LIKE MANY CASTLES, HIMEJI HAS A MOAT

SURROUNDING THE CASTLE GROUNDS.

SECOND, THERE IS A HIGH WALL BEHIND THE MOAT

ENCLOSING THE CASTLE GROUNDS.

THERE IS A BRIDGE NEAR THE MAIN ENTRANCE

AND AN INVADING FORCE WOULD MOVE IN THIS DIRECTION.

BUT ONCE INSIDE AND PAST THE MAIN GATE

THE GROUNDS BECOME A HUGE MAZE WITH VARIOUS PATHWAYS

LEADING TO DEAD ENDS.

THE LONG WINDING PATH THROUGH THIS MAZE

IS ACTUALLY A SPIRAL PATH TO THE MAIN TOWER.

MOVING ALONG A SPIRAL PATH MEANS THAT YOU

TURN 360 DEGREES OR ONE FULL CIRCUIT

FROM YOUR INITIAL POSITION.

YOU CAN SEE THIS ON THE NSPIRE.

ACTIVATE THE SEGMENT TOOL.

CONSTRUCT AN ANGLE.

FIRST CREATE ONE SIDE OF THE ANGLE,

PRESSING ENTER FOR EACH END POINT.

THEN PRESS ENTER ONCE AGAIN OVER ONE OF THE ENDPOINTS.

CREATE A SECOND SEGMENT

IN THE PROCESS CREATING THE ANGLE.

NOW MEASURE THE ANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

AFTER YOU HAVE MEASURED THE ANGLE

PRESS ESCAPE AND MOVE THE POINTER

TO ONE OF THE ENDPOINTS AS SHOWN.

HOLD THE CLICK KEY UNTIL YOU SELECT THE POINT.

MOVE THE POINT SO THAT THE ANGLE MEASURE IS 0 DEGREES.

NEXT, MOVE THE POINT UNTIL THE MEASURE IS 180 DEGREES.

WHAT DO YOU NOTICE?

IT'S A STRAIGHT LINE.

SO IF YOU WERE TO CONTINUE ROTATING THE POINT

SO THAT IT IS BACK TO WHERE YOU STARTED

THIS WOULD RESULT IN ANOTHER 180 DEGREES

FOR A TOTAL OF 360 DEGREES.

AN INVADING ARMY WOULD NOT REALIZE THAT IT IS GOING

IN CIRCLES SINCE THE SPIRAL PATH IS GRADUALLY CIRCULAR.

IN ADDITION, ON ITS WAY THROUGH THE MAZE

THE PATH GRADUALLY BECOMES STEEPER.

THAT PATH HAS A SLOPE OF ABOUT TWO DEGREES.

THIS IS AN ACUTE ANGLE SINCE IT IS LESS THAN 90 DEGREES.

TRY TO MODEL A TWO DEGREE ANGLE ON THE NSPIRE.

AS YOU CAN SEE, SUCH A SLOPE IS GRADUAL ENOUGH

THAT AN ARMY WOULD NEED TO TRAVEL FARTHER AND

EXERT ITSELF MORE TO REACH THE BASE OF THE TOWER.

FINALLY UPON REACHING THE BASE OF THE TOWER

THE ARMY WOULD BE STARING AT THIS IMPOSING STRUCTURE.

SEEN FROM THE SIDE, THE MAIN TOWER

MAKES AN OBTUSE ANGLE WITH THE GROUND

OR AN ANGLE GREATER THAN 90 DEGREES.

ALSO NOTICE THAT THE LINE OF SIGHT FROM GROUND LEVEL

PREVENTS AN ATTACKER FROM GETTING A VIEW OF

THE TOP OF THE TOWER GIVING THE CASTLE'S DEFENDERS

STILL ONE MORE ELEMENT OF SURPRISE

IN ATTACKING THE ARMY AT THE BASE OF THE TOWER.

HIMEJI CASTLE'S DEFENSES OWE A GREAT DEAL TO THE WAY

IN WHICH ANGLES ARE USED, FROM THE SLOPING PATHS

WHERE INVADERS ARE SLOWED DOWN,

TO THE BLIND CORNERS OF ITS BYZANTINE MAZE

WHERE DEFENDERS COULD AMBUSH THE INVADERS,

TO THE INSURMOUNTABLE WALLS OF THE MAIN TOWER.

THE PROPERTIES OF ANGLES ARE USED FOR

DEFENSIVE PURPOSES AND YET HIMEJI REMAINS

AN ELEGANT PCE OF ARCHITECTURE THAT ADDS

MAJESTY AND SERENITY TO ITS SURROUNDINGS.