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X
- THE AREA OF AN EQUILATERAL TRIANGLE
IS 177 SQUARE CENTIMETERS.
AN EQUILATERAL TRIANGLE HAS 3 SIDES OF THE SAME LENGTH
AND 3 ANGLES OF THE SAME MEASURE.
GIVEN THIS INFORMATION,
WE WANT TO FIND THE PERIMETER OF THE TRIANGLE
AND ROUND TO 2 DECIMAL PLACES AS NEEDED.
TO DO THIS PROBLEM, WE'RE NOT GOING TO USE TRIGONOMETRY.
WE'RE GOING TO USE THE PYTHAGOREAN THEOREM
AND THE AREA FORMULA FOR A TRIANGLE.
SO FOR CONVENIENCE, WE'RE GOING TO LET THE LENGTH
OF EACH SIDE BE 2X CENTIMETERS.
SO THIS WILL BE 2X,
THIS WILL BE 2X,
AND THIS WILL BE 2X.
WHAT WE WANT TO DO IS SET UP AN EQUATION
INVOLVING THE AREA OF THE TRIANGLE THAT ONLY INVOLVES X.
TO DO THIS, WE'RE GOING TO USE THIS RIGHT TRIANGLE HERE
AND EXPRESS THE HEIGHT OF THIS TRIANGLE IN TERMS OF X.
BUT BEFORE WE DO THIS, WE NEED TO RECOGNIZE A COUPLE OF THINGS.
FIRST, IF THE LENGTH OF THIS SIDE IS 2X,
THE HEIGHT IS GOING TO BISECT THIS SIDE
SO THE LENGTH OF THIS SIDE HERE WOULD BE X,
AND THEN WE'LL CALL THIS H FOR HEIGHT.
AGAIN, OUR GOAL IS TO EXPRESS THE HEIGHT IN TERMS OF X,
AND WE'LL DO THIS BY USING THE PYTHAGOREAN THEOREM
EXPRESSED HERE.
SO USING THE PYTHAGOREAN THEOREM,
WE CAN WRITE EQUATION H SQUARED + X SQUARED
MUST EQUAL 2X SQUARED.
SO WE HAVE H SQUARED + X SQUARED EQUALS,
THIS WILL BE 4X SQUARED,
SO H SQUARED IS GOING TO BE EQUAL TO,
SUBTRACT X SQUARED ON BOTH SIDES, WE WOULD HAVE 3X SQUARED.
AND NOW, WE'LL TAKE THE SQUARE ROOT
OF BOTH SIDES OF THE EQUATION.
AND SINCE H IS A LENGTH,
WE'RE ONLY CONCERNED ABOUT THE POSITIVE SQUARE ROOT.
SO H IS GOING TO BE EQUAL TO,
THE SQUARE ROOT OF X SQUARED WOULD BE X,
SO SIMPLIFIES TO X SQUARE ROOT OF 3.
AND NOW, WE CAN SET UP AN EQUATION TO SOLVE FOR X
BY USING WHAT WE KNOW ABOUT THE AREA.
THE AREA OF THE TRIANGLE = 177 SQUARE CENTIMETERS.
WELL, THE AREA OF THE TRIANGLE = 1/2 x THE BASE x THE HEIGHT.
SO FOR THIS SITUATION, WE'RE GOING TO HAVE 1/2 x THE BASE,
WHICH HAS A LENGTH OF 2X x THE HEIGHT EXPRESSED IN TERMS OF X
IS GOING TO BE X SQUARE ROOT OF 3,
AND THIS MUST EQUAL 177 SQUARE CENTIMETERS.
SO NOW, WE'RE GOING TO TAKE THIS EQUATION HERE AND SOLVE FOR X.
THE FIRST THING WE SHOULD NOTICE IS THAT
THIS 2 AND THIS 2 SIMPLIFY OUT,
SO IT SIMPLIFIES TO THE EQUATION X SQUARED x SQUARE ROOT OF 3
IS EQUAL TO 177.
LET'S GO AHEAD AND TAKE THIS ON TO THE NEXT PAGE.
WE'LL GO AHEAD AND DIVIDE BOTH SIDES BY SQUARE ROOT OF 3,
SO WE HAVE X SQUARED = 177 DIVIDED BY SQUARE ROOT OF 3.
AND NOW, WE'LL TAKE THE SQUARE ROOT
OF BOTH SIDES OF THE EQUATION,
AND AGAIN WE'RE ONLY CONCERNED ABOUT THE POSITIVE SQUARE ROOT
OR PRINCIPAL SQUARE ROOT HERE.
SO WE GET A DECIMAL APPROXIMATION HERE FOR X,
AND THEN WE'LL GO BACK AND FINALLY ANSWER THE QUESTION
ABOUT WHAT'S THE PERIMETER OF THE TRIANGLE.
SO WE'RE GOING TO HAVE
THE SQUARE ROOT OF THIS FRACTION HERE,
WHICH IS 177 DIVIDED BY THE SQUARE ROOT OF 3,
SO WE HAVE A PARENTHESIS HERE FOR THE SQUARE ROOT OF 3,
ANOTHER PARENTHESIS HERE FOR THE OUTER SQUARE ROOT,
SO X IS APPROXIMATELY 10.11.
AND AGAIN, THIS IS CENTIMETERS.
NOW, LET'S GO BACK TO THE PREVIOUS SCREEN FOR A MOMENT.
NOTICE THE PERIMETER OF THIS TRIANGLE
WITH A DISTANCE AROUND THE OUTSIDE
WOULD BE 2X + 2X + 2X OR 6X.
AGAIN, THE LENGTH OF EACH SIDE IS 2X CENTIMETERS,
SO THE PERIMETER = 6X
WHICH WOULD BE APPROXIMATELY = 6 x 10.11 CENTIMETERS,
WHICH IS GOING TO BE 60.66 CENTIMETERS.
OKAY, I HOPE THIS EXPLANATION HELPS.