Proof: Problem 1 Writing a Coordinate Proof

Use coordinate geometry to prove that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

Given: Δ OEF cap delta  is a right triangle.

M is the midpoint of E F ¯ e f bar

Prove: E M = e m equals   F M = f m equals  OM

Coordinate Proof:

By the Midpoint Formula, M = ( 2 a + 0 2 , 0 + 2 b 2 ) = ( a , b ) . m equals open . fraction 2 eh plus 0 , over 2 end fraction . comma . fraction 0 plus 2 b , over 2 end fraction . close equals open eh comma b close .

By the Distance Formula,

O M = a 2 + b 2     F M = ( 2 a − a ) 2 + ( 0 − b ) 2 E M = ( 0 − a ) 2 + ( 2 b − b ) 2   = a 2 + b 2   = a 2 + b 2 table with 3 rows and 4 columns , row1 column 1 , o m , column 2 equals . square root of eh squared , plus , b squared end root , column 3 , column 4 , row2 column 1 , f m , column 2 equals . square root of open 2 eh minus eh close squared . plus . open 0 minus b close squared end root , column 3 e m , column 4 equals . square root of open 0 minus eh close squared . plus . open 2 b minus b close squared end root , row3 column 1 , , column 2 equals . square root of eh squared , plus , b squared end root , column 3 , column 4 equals . square root of eh squared , plus , b squared end root , end table

Since E M = F M = O M , e m equals f m equals o m comma  the midpoint of the hypotenuse is equidistant from the vertices of the right triangle.

✓ Got It?

1. Reasoning What is the advantage of using coordinates O(0, 0), E(0, 2b), and F(2a, 0) rather than O(0, 0), E(0, b), and F(a, 0)?