- WELCOME TO EXAMPLES OF SOLVING 30-60-90 TRIANGLES,
SOMETIMES CALLED 30-60 RIGHT TRIANGLES.
TAKE A LOOK AT THIS DIAGRAM ON THE LEFT.
NORMALLY WHEN WE HAVE A 30-60-90 TRIANGLE
WE LABEL THE SIDE OPPOSITE 30 DEGREE 1, THE HYPOTENUSE 2,
AND THE SIDE OPPOSITE THE 60 DEGREE ANGLE SQUARE ROOT 3
BECAUSE THE RATIO AMONG THE THREE SIDES STAYS CONSTANT.
IT DOESN'T MEAN THAT ANY 30-60-90 TRIANGLE
IS GOING TO HAVE THE LENGTHS AS LABELED HERE.
IT COULD BE ANY MULTIPLE OF THESE.
SO IF WE TAKE EACH OF THESE THREE SIDES
AND MULTIPLY IT BY X, 1 x X WOULD BE X, 2 x X WOULD BE 2X,
AND SQUARE ROOT 3 x X WOULD BE X SQUARE ROOT 3.
THIS IS ACTUALLY THE DIAGRAM WE WANT TO USE TO SOLVE
ANY 30-60-90 TRIANGLE
IF WE KNOW THE LENGTH OF ONE OF THE SIDES.
LET'S TAKE A LOOK AT OUR EXAMPLES.
HERE WE WANT TO SOLVE THIS 30-60-90 TRIANGLE,
AND THE SIDE OPPOSITE THE 30 DEGREE ANGLE IS 5 FEET.
LOOKING AT THE DIAGRAM FROM THE PREVIOUS PAGE,
THE SIDE OPPOSITE THE 30 DEGREE ANGLE IS EQUAL TO X.
SO FOR THIS PROBLEM X IS EQUAL TO 5 FEET,
WHICH MEANS THE LENGTH OF THE HYPOTENUSE WOULD BE 2 x X.
WELL, 2 x 5 FEET WOULD BE 10 FEET,
SO THE LENGTH OF THE HYPOTENUSE IS 10 FEET.
AND THE LENGTH OF THE SIDE
THAT'S OPPOSITE THE 60 DEGREE ANGLE IS X x SQUARE ROOT 3.
SO THIS LEG HERE WOULD HAVE A LENGTH OF 5 x SQUARE ROOT 3.
LET'S TAKE A LOOK AT A COUPLE MORE EXAMPLES.
THEY'RE NOT ALL QUITE THIS STRAIGHT FORWARD.
SO, AGAIN, HERE'S THE DIAGRAM FROM THE FIRST PAGE,
AND WE WANT TO SOLVE THIS TRIANGLE HERE.
SO NOTICE HOW WE'RE GIVEN THAT THE SIDE OPPOSITE
THE 60 DEGREE ANGLE IS EQUAL TO 9 CENTIMETERS.
WHICH FROM OUR DIAGRAM HERE,
MUST BE EQUAL TO X SQUARE ROOT 3.
SO IF WE CAN SOLVE FOR X,
WE CAN DETERMINE THE LENGTH OF THE OTHER TWO SIDES.
SO THIS TIME WE'LL SET IT UP AS AN EQUATION
X x SQUARE ROOT 3 MUST EQUAL 9 CENTIMETERS.
LET'S GO AND JUST WRITE IT AS 9.
SO WE'LL DIVIDE BY SQUARE ROOT 3,
X = 9 DIVIDED BY SQUARE ROOT 3.
LET'S GO AHEAD AND RATIONALIZE THIS.
SO X IS GOING TO BE EQUAL TO 9 SQUARE ROOT 3 DIVIDED BY 3,
WHICH WILL GIVE US 3 SQUARE ROOTS OF 3.
SO X WILL BE THE LENGTH OF THE SIDE OPPOSITE
THE 30 DEGREE ANGLE.
SO THIS WILL BE 3 SQUARE ROOT 3 CENTIMETERS.
AND THE HYPOTENUSE WILL HAVE A LENGTH OF 2 x X.
WELL, 2 x 3 SQUARE ROOT 3 IS EQUAL TO 6 SQUARE ROOTS OF 3.
SO THE HYPOTENUSE HAS A LENGTH
OF 6 SQUARE ROOT 3 CENTIMETERS.
NOW LETS TAKE A LOOK AT ONE OTHER EXAMPLE.
AGAIN, NOTICE WE HAVE A 30-60-90 TRIANGLE.
BUT IN THIS PROBLEM,
WE'RE GIVEN THE LENGTH OF THE HYPOTENUSE,
AND HYPOTENUSE IS EQUAL TO 2 x X.
SO 2X MUST EQUAL 4 METERS, DIVIDE BOTH SIDES BY 2.
NOW WE HAVE X = 2 METERS.
THIS IS ALL WE NEED TO DETERMINE
THE LENGTH OF THE OTHER TWO SIDES.
AGAIN, THE SIDE OPPOSITE THE 30 DEGREE ANGLE IS EQUAL TO X,
SO THIS WILL BE 2 METERS.
AND THE SIDE OPPOSITE THE 60 DEGREE ANGLE
IS EQUAL TO X x SQUARE ROOT 3.
SO IF X IS 2, WE HAVE 2 SQUARE ROOT 3,
AND THIS WOULD BE METERS.
SO THE KEY TO THESE TYPES OF PROBLEMS
IS TAKING THE GIVEN INFORMATION
AND SETTING UP AN EQUATION TO SOLVE FOR X.
AND THEN ONCE WE HAVE X,
WE CAN DETERMINE THE LENGTH OF THE REMAINING SIDES.