Proof Problem 2 Writing a Coordinate Proof

Write a coordinate proof of the Trapezoid Midsegment Theorem.

Given: M N ¯ m n bar  is the midsegment of trapezoid ORAP.

Prove: M N ¯ ∥ O P ¯ , M N ¯ ∥ R A ¯ , M N = 1 2 ( O P + R A ) m n bar , parallel to , o p bar , comma , m n bar , parallel to , r eh bar , comma . m n equals , 1 half , open o p plus r eh close

Coordinate Proof:

Use the Midpoint Formula to find the coordinates of M and N.

M = ( 2 b + 0 2 , 2 c + 0 2 ) = ( b , c ) N = ( 2 a + 2 d 2 , 0 + 2 c 2 ) = ( a + d , c ) table with 2 rows and 1 column , row1 column 1 , m equals open . fraction 2 b plus 0 , over 2 end fraction . comma . fraction 2 c plus 0 , over 2 end fraction . close equals open b comma c close , row2 column 1 , n equals open . fraction 2 eh plus 2 d , over 2 end fraction . comma . fraction 0 plus 2 c , over 2 end fraction . close equals open eh plus d comma c close , end table

Use the Slope Formula to determine whether M N ¯ m n bar  is parallel to O P ¯ o p bar  and R A ¯ . r eh bar , .

slope of M N ¯ = c − c ( a + d ) − b = 0 slope of R A ¯ = 2 c − 2 c 2 d − 2 b = 0 slope of O P ¯ = 0 − 0 2 a − 0 = 0 table with 3 rows and 1 column , row1 column 1 , slopeof . m n bar , equals . fraction c minus c , over open eh plus d close minus b end fraction . equals 0 , row2 column 1 , slopeof . r eh bar , equals . fraction 2 c minus 2 c , over 2 d minus 2 b end fraction . equals 0 , row3 column 1 , slopeof . o p bar , equals . fraction 0 minus 0 , over 2 eh minus 0 end fraction . equals 0 , end table

The three slopes are equal, so M N ¯ ∥ O P ¯ m n bar , parallel to , o p bar  and M N ¯ ∥ R A ¯ . m n bar , parallel to , r eh bar , .

Use the Distance Formula to find and compare MN, OP, and RA.

M N = [ ( a + d ) − b ) ] 2 + ( c − c ) 2 = a + d − b O P = ( 2 a − 0 ) 2 + ( 0 − 0 ) 2 = 2 a R A = ( 2 d − 2 b ) 2 + ( 2 c − 2 c ) 2 = 2 d − 2 b M N = ? 1 2 ( O P + R A ) Check that  M N = 1 2 ( O P + R A )  is true. a + b − d = ? 1 2 [ 2 a + ( 2 d − 2 b ) ] Substitute. a + d − b = a + d − b   ✓ Simplify. table with 4 rows and 1 column , row1 column 1 , m n equals . square root of left bracket open eh plus d close minus b close , right bracket squared , plus . open c minus c close squared end root . equals eh plus d minus b , row2 column 1 , o p equals . square root of open 2 eh minus 0 close squared . plus . open 0 minus 0 close squared end root . equals 2 eh , row3 column 1 , r eh equals . square root of open 2 d minus 2 b close squared . plus . open 2 c minus 2 c close squared end root . equals 2 d minus 2 b , row4 column 1 , table with 3 rows and 3 columns , row1 column 1 , m n , column 2 modified equals with question mark above , 1 half , open o p plus r eh close , column 3 cap checkthat m n equals , 1 half , open o p plus r eh close . istrue. , row2 column 1 , eh plus b minus d , column 2 modified equals with question mark above , 1 half , left bracket 2 eh plus open 2 d minus 2 b close right bracket , column 3 cap substitute. , row3 column 1 , eh plus d minus b , column 2 equals eh plus d minus b check mark , column 3 cap simplify. , end table , end table

So, (1) the midsegment of a trapezoid is parallel to its bases, and

(2) the length of the midsegment of a trapezoid is half the sum of the lengths of the bases.

✓ Got It?

2. Write a coordinate proof of the Triangle Midsegment Theorem (Theorem 5-1).