Here's Why It Works You can verify that the Triangle Midsegment Theorem works for a particular triangle. Use the following steps to show that D E ¯ ║ A B ¯ d e bar , box drawings double vertical , eh b bar  and that D E = 1 2 A B d e equals , 1 half eh b  for a triangle with vertices at A(4, 6), B(6, 0), and C(0, 0), where D and E are the midpoints of C A ¯ c eh bar  and C B ¯ . c b bar , .

• Step 1 Use the Midpoint Formula, M = ( x 1 + x 2 2 , y 1 + y 2 2 ) , m equals . open . fraction x sub 1 , plus , x sub 2 , over 2 end fraction . comma . fraction y sub 1 , plus , y sub 2 , over 2 end fraction . close . comma

  to find the coordinates of D and E.

  The midpoint of C A ¯ c eh bar  is D ( 0 + 4 2 , 0 + 6 2 ) = D ( 2 , 3 ) . d . open . fraction 0 plus 4 , over 2 end fraction . comma . fraction 0 plus 6 , over 2 end fraction . close . equals d . open , 2 comma 3 , close . .

  The midpoint of C B ¯ c b bar  is E ( 0 + 6 2 , 0 + 0 2 ) = E ( 3 , 0 ) . e . open . fraction 0 plus 6 , over 2 end fraction . comma . fraction 0 plus 0 , over 2 end fraction . close . equals e . open , 3 comma 0 , close . .

• Step 2 To show that the midsegment D E ¯ d e bar  is parallel to the side A B ¯ , eh b bar , comma  find the slope, m = y 2 − y 1 x 2 − x 1 , m equals . fraction y sub 2 , minus , y sub 1 , over x sub 2 , minus , x sub 1 end fraction . comma  of each segment.

  slope of  D E ¯ = 0 − 3 3 − 2   = − 3 1   = − 3 slope of  A B = 0 − 6 6 − 4   = − 6 2   = − 3 table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 2 columns , row1 column 1 , slopeof . d e bar , column 2 equals . fraction 0 minus 3 , over 3 minus 2 end fraction , row2 column 1 , , column 2 equals , negative 3 over 1 , row3 column 1 , , column 2 equals negative 3 , end table , column 2 table with 3 rows and 2 columns , row1 column 1 , slopeof eh b , column 2 equals . fraction 0 minus 6 , over 6 minus 4 end fraction , row2 column 1 , , column 2 equals , negative 6 over 2 , row3 column 1 , , column 2 equals negative 3 , end table , end table

• Step 3 To show D E = 1 2 A B , d e equals , 1 half eh b comma  use the Distance Formula, d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d equals . square root of open . x sub 2 , minus , x sub 1 . close squared . plus . open . y sub 2 , minus , y sub 1 . close squared end root  to find DE and AB.

  D E = ( 3 − 2 ) 2 + ( 0 − 3 ) 2   = 1 + 9   = 10 A B = ( 6 − 4 ) 2 + ( 0 − 6 ) 2   = 4 + 36   = 40   = 2 10 table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 2 columns , row1 column 1 , d e , column 2 equals . square root of open 3 minus 2 close squared . plus . open 0 minus 3 close squared end root , row2 column 1 , , column 2 equals , square root of 1 plus 9 end root , row3 column 1 , , column 2 equals square root of 10 , end table , column 2 table with 4 rows and 2 columns , row1 column 1 , eh b , column 2 equals . square root of open 6 minus 4 close squared . plus . open 0 minus 6 close squared end root , row2 column 1 , , column 2 equals , square root of 4 plus 36 end root , row3 column 1 , , column 2 equals square root of 40 , row4 column 1 , , column 2 equals 2 square root of 10 , end table , end table

Since 10 = 1 2 ( 2 10 ) , square root of 10 equals , 1 half . open 2 square root of 10 close comma  you know that D E = 1 2 A B . d e equals , 1 half eh b .

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