[ MUSIC ]
- WELCOME TO PARALLEL AND PERPENDICULAR LINES.
THE GOALS OF THIS VIDEO ARE TO DETERMINE IF TWO LINES
ARE PARALLEL OR PERPENDICULAR
AND ALSO TO DETERMINE THE EQUATION OF LINES
THAT ARE PARALLEL OR PERPENDICULAR.
PARALLEL LINES ARE TWO LINES THAT NEVER INTERSECT.
TWO LINES ARE PARALLEL IF THEY HAVE THE SAME SLOPE
AND DIFFERENT Y INTERCEPTS.
SO IF THEY'RE BOTH IN SLOPE INTERCEPT FORM M1 MUST EQUAL M2
AND B1 CANNOT EQUAL B2.
LET'S EXPLORE THIS IDEA WITH AN APPLET.
IF YOU WANT TO TRY IT YOURSELF
I WOULD ENCOURAGE YOU TO VISIT THE WEBSITE.
SO HERE WE HAVE A LINE.
LET'S GO AHEAD AND PLACE THESE POINTS
SO WE CAN EASILY DETERMINE THE SLOPE.
THE SLOPE WOULD BE -2.
TO GO FROM POINT "A" TO POINT B WE'D GO DOWN TWO AND RIGHT ONE.
NOW IF WE INTRODUCE A SECOND LINE LETS PUT THESE TWO POINTS
SO THAT THE SLOPE IS ALSO -2.
SO I PUT THE FIRST POINT HERE AND THE SECOND POINT HERE.
AGAIN, IF WE GO DOWN TWO AND RIGHT ONE WE'RE ON THIS RED LINE
AND YOU CAN SEE THOSE LINES ARE PARALLEL.
THEY CONTINUE ON FOREVER AND THEY WOULD NEVER INTERSECT.
NOW, AS SOON AS I CHANGE THE SLOPES THEY'RE NOT THE SAME,
OF COURSE THEY INTERCEPT.
AGAIN, IF I PLACE THEM BACK WHERE THE SLOPE IS THE SAME
THEY'RE PARALLEL
BUT NOTICE THAT THIS ONE CROSSES THE Y-AXIS AT APPROXIMATELY ONE,
THIS ONE CROSSES AT APPROXIMATELY SEVEN.
THE REASON THE Y INTERCEPTS MUST BE DIFFERENT
BECAUSE IF THEY WERE THE SAME
OF COURSE THE LINES WOULD BE ON TOP OF EACH OTHER.
SO THE SLOPES MUST BE THE SAME,
THE Y INTERCEPTS MUST BE DIFFERENT.
AGAIN, HERE'S A PROBABLY SEE A LITTLE BIT BETTER.
YOU CAN SEE THESE LINES ARE PARALLEL.
THEY WOULD NEVER INTERSECT.
AND IF WE CHECK THE SLOPES
WE BASICALLY HAVE TO FIND TWO POINTS,
FIND THE RISE OVER THE RUN.
IF I GO UP ONE AND RIGHT TWO THIS SLOPE WOULD BE 1/2.
AND OF COURSE ON THE RED LINE IF I GO UP ONE AND RIGHT TWO
THE SLOPE WAS ALSO 1/2.
PERPENDICULAR LINES ARE LINES THAT INTERSECT
AND FORM A RIGHT ANGLE OR A 90 DEGREE ANGLE.
TWO LINES ARE PERPENDICULAR
IF THE PRODUCT OF THE SLOPES IS -1
OR IF THE SLOPES ARE NEGATIVE RECIPROCALS.
THE ONLY EXCEPTION TO THIS RULE IS IF ONE LINE IS VERTICAL
AND THE OTHER IS HORIZONTAL,
THEY'RE ALSO PERPENDICULAR.
SO HERE'S WHAT WE MEAN.
IF THE LINES ARE IN SLOP INTERCEPT FORM
THEY'RE PERPENDICULAR IF M1 x M2 = -1
OR IF THEY'RE NEGATIVE RECIPROCALS.
LET'S VIEW AN EXAMPLE OF THIS.
IF I ASK TO FIND THE SLOPE OF THE BLUE LINE
PICKING THESE TWO POINTS, IF I GO UP ONE AND RIGHT TWO,
THE SLOPE OF THE BLUE LINE IS + 1/2.
IF WE TRY TO FIND THE SLOPE OF THE RED LINE
USING THESE TWO POINTS
WE WOULD GO DOWN TWO AND RIGHT ONE.
WELL, DOWN TWO AND RIGHT ONE GIVES US A SLOPE OF -2/1.
THESE ARE NEGATIVE RECIPROCALS OF ONE ANOTHER.
BASICALLY YOU FLIP IT OVER AND CHANGE THE SIGN.
THEREFORE, THEY MUST BE PERPENDICULAR LINES
OR INTERSECT AT A RIGHT ANGLE.
THE OTHER CHECK IS TO FIND THEIR PRODUCT.
+1/2 x -2/1 OF COURSE IS -2/2 WHICH IS EQUAL TO -1.
THAT'S ANOTHER WAY TO CHECK TO SEE IF LINES ARE PERPENDICULAR.
LET'S GO AND LOOK AT AN EXAMPLE.
DETERMINE IF THESE LINES ARE PARALLEL, PERPENDICULAR,
OR NEITHER.
SO LETS START BY COMPARING THE SLOPES.
WELL, LUCKILY THIS FIRST LINE IS SLOPE INTERCEPT FORM.
SO WE'LL SAY THAT M1 IS EQUAL TO 2/3.
IN ORDER TO FIND THE SLOPE OF THIS LINE
WE'RE GOING TO HAVE TO SOLVE IT FOR Y
OR PUT IT IN Y = MX + B FORM.
SO LET'S GO AHEAD AND DO THAT.
SO LET'S ADD THE X TERM TO THE OTHER SIDE.
SO WE HAVE 3Y = 2X + 3. LET'S DIVIDE BY 3.
WE CAN GO AHEAD AND STOP HERE
AND DETERMINE THAT THE SLOPE OF THE SECOND LINE IS ALSO 2/3.
NOW, REMEMBER THAT'S NOT ENOUGH INFORMATION
TO DETERMINE THAT THEY'RE PARALLEL.
WE ALSO HAVE TO COMPARE THE Y INTERCEPTS.
THIS Y INTERCEPT IS EQUAL TO -4,
THIS Y INTERCEPT WOULD SIMPLIFY TO +1.
SO WE HAVE THE SAME SLOPES, DIFFERENT Y INTERCEPTS,
THEREFORE WE CAN CONCLUDE THAT THE TWO LINES ARE PARALLEL.
IF WE WANTED TO, WE COULD GRAPH IT TO VERIFY.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
SAME QUESTION, DIFFERENT EQUATIONS,
SO WE HAVE TO SOLVE BOTH EQUATIONS FOR Y
TO PUT THEM IN SLOPE INTERCEPT FORM.
SO LET'S TAKE OUR FIRST LINE,
SUBTRACT 2X ON BOTH SIDES, AND DIVIDE BY +5.
SO WE HAVE Y = -2/5X + 9/5.
SO LET'S GO AHEAD AND RECORD OUR INFORMATION.
WE KNOW THE SLOPE IS EQUAL TO -2/5
AND THE Y INTERCEPT IS EQUAL TO 9/5.
LET'S SOLVE THE SECOND EQUATION FOR Y.
SO WE'LL SUBTRACT 10X ON BOTH SIDES.
SO -4Y = -10X + 12. WE'LL DIVIDE BY -4.
SO THIS SIMPLIFIES TO A +5/2X - 3.
SO WE HAVE A SLOPE OF 5/2 AND A Y INTERCEPT OF -3.
WELL, THE SLOPES AREN'T THE SAME SO THEY CAN'T BE PARALLEL.
IF THEY'RE PERPENDICULAR
THE SLOPES MUST BE NEGATIVE RECIPROCALS.
+5/2, -2/5, THOSE ARE NEGATIVE RECIPROCALS.
OR IF WE WANTED TO FIND THEIR PRODUCT
WE'D HAVE -10/10 OF COURSE EQUAL TO -1.
SO THESE LINES ARE PERPENDICULAR.
NOW WE'LL TAKE A LOOK AT THIS FROM A DIFFERENT DIRECTION.
DETERMINE THE EQUATION OF THE LINE THAT IS PARALLEL
TO X - 3Y = 9 AND PASSES THROUGH THE POINT (6,2).
WELL, IF THE NEW LINE IS PARALLEL TO THIS LINE
WE NEED TO START BY FINDING THE SLOPE OF THIS LINE.
SO WE'LL SOLVE THIS FOR Y, WE'LL SUBTRACT X ON BOTH SIDES,
DIVIDE BY -3.
SO WE HAVE Y IS EQUAL TO-- THIS WOULD BE -1/-3 OR 1/3X - 3.
SO THE SLOPE OF THIS LINE IS EQUAL TO 1/3.
WELL, IF OUR LINE IS PARALLEL TO THIS
OUR SLOPE MUST ALSO EQUAL 1/3
AND IT MUST PASS THROUGH THE POINT (6,2).
SO IF WE'RE TRYING TO FIND THE EQUATION OF THIS LINE
IN SLOPE INTERCEPT FORM
WE ALREADY KNOW THE SLOPE.
SO REMEMBER IN ORDER TO FIND THE Y INTERCEPT
WE CAN REPLACE X WITH 6 AND Y WITH 2.
SINCE THIS POINT IS ON OUR LINE THAT'LL ALLOW US TO SOLVE FOR B.
LOOKS LIKE OUR VALUE FOR THE Y INTERCEPT IS ZERO.
THEREFORE, THE EQUATION OF THE LINE THAT WE'RE LOOKING FOR
IS Y = 1/3X.
AGAIN, I WOULD ENCOURAGE YOU TO GRAPH THESE TWO LINES TO VERIFY.
BUT, AGAIN, WE CAN ALREADY SEE THAT OUR SLOPES ARE THE SAME,
OUR Y INTERCEPTS ARE DIFFERENT, SO IT SHOULD BE PARALLEL.
LET'S TAKE A LOOK AT ONE MORE.
DETERMINE THE EQUATION OF A LINE THAT IS PERPENDICULAR TO Y = 4
AND PASSES THROUGH THE POINT (-3,-2).
SO THIS ONE DOESN'T FIT THE NORMAL PATTERN
OF WHAT WE'RE USED TO.
SO LET'S GO AHEAD AND GRAPH SOME OF THIS INFORMATION
ON THE COORDINATE PLAIN.
WE KNOW OUR LINE IS PERPENDICULAR TO Y = 4.
WE KNOW THAT Y = 4 IS A HORIZONTAL LINE.
IT'LL LOOK SOMETHING LIKE THIS.
SO RIGHT AWAY WE SHOULD BE ABLE TO RECOGNIZE IF OUR LINE
IS GOING TO BE PERPENDICULAR TO THIS IT MUST BE A VERTICAL LINE.
NOW LET'S ALSO PLOT OUR POINT (-3,-2).
THAT WOULD BE THE POINT HERE.
SO THE QUESTION REALLY IS WHAT IS THE EQUATION OF THE LINE
THAT IS VERTICAL
AND PASSES THROUGH THAT GIVEN POINT?
WELL, IF YOU REMEMBER THE EQUATIONS OF HORIZONTAL
AND VERTICAL LINES,
Y EQUALS SOME CONSTANT IS ALWAYS A HORIZONTAL LINE
AND X EQUALS SOME CONSTANT IS ALWAYS A VERTICAL LINE.
SO IF THIS LINE PASSES THROUGH THE POINT (-3,-2)
THE EQUATION OF THIS VERTICAL LINE MUST BE X = -3.
ANY POINT ON THIS VERTICAL LINE WILL HAVE AN X COORDINATE OF -3.
I HOPE THIS SHORT VIDEO HELPED EXPLAIN THE PROPERTIES
OF HORIZONTAL AND VERTICAL LINES.
THANK YOU FOR WATCHING AND HAVE A GOOD DAY.