If a right triangle’s side lengths are #a, b, and c, and b−a=2, c−b=2#, what is the area minus the perimeter?

1 Answer
Alan P.
Oct 29, 2017

#"Area"_triangle - "Perimeter"_triangle = 0#

Explanation:

For a right triangle with sides #a, b," and " c#
#color(white)("xx"){:("If",b-a=2,rarr,b=a+2,,), ("If",c-b=2,rarr,c=b+2,rarr,c=a+4) :}#

Since #c > b > a#, the hypotenuse must be of length #c#

#a^2+b^2=c^2#

#a^2+(a+2)^2=(a+4)^2#

#a^2+a^2+4a+4=a^2+8a+16#

#2a^2+4a+4=a^2+8a+16#

#a^2-4a-12=0#

#(a-6)(a+2)=0#

#{: (rarr,(a-6)=0," or ",(a+2)=0), (,rarr a=6,,rarr a=-2), (,,,"impossible") :}#

Therefore
#color(white)("XXX")a=6#,
#color(white)("XXX")b=8#, and
#color(white)("XXX")c=12#

#"Perimeter"_triangle =6+8+12=24#

#"Area"_triangle = (6xx8)/2=24#

#"Area"_triangle - "Perimeter"_triangle = 24-24=0#

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