SOLUTION: MON is a triangle such that MO= x+3, MN= 3 and ON= 6-x. 1) for what values of x do the lengths MO and ON exist? Justify 2) calculate x such that triangle MON is inscribed in a ci

Question 1037091: MON is a triangle such that MO= x+3, MN= 3 and ON= 6-x.
1) for what values of x do the lengths MO and ON exist? Justify
2) calculate x such that triangle MON is inscribed in a circle
of diameter the side [ON]

Answer by Edwin McCravy(20066) (Show Source): You can put this solution on YOUR website!
MON is a triangle such that MO = x+3, MN = 3 and ON = 6-x.
1) for what values of x do the lengths MO and ON exist? Justify

The sum of any two sides of a triangle must be
strictly greater than the third side:

  MO+MN > ON          MO+ON > MN     MN+ON > MO 
x+3 + 3 > 6-x     x+3 + 6-x > 3      3+6-x > x+3    
    x+6 > 6-x             9 > 3        9-x > x+3 
     2x > 0                            -2x > -6 
      x > 0                              x < 3

So from those we get that 0 < x < 3

2) calculate x such that triangle MON is inscribed in a circle
of diameter the side [ON]

If a triangle inscribed in a circle has a side which is a 
diameter, then the triangle is a right angle.

We use the converse of the Pythagorean theorem, which states
that if the sides of a triangle a,b,c are such that a2+b2=c2,
then the triangle is a right triangle.

    MO2 + MN2 = ON2
  (x+3)2 + 32 = (6-x)2
  x2+6x+9 + 9 = 36-12x+x2
     x2+6x+18 = 36-12x+x2
          18x = 18
            x = 1

MO = x+3 = 1+3 = 4 
MN = 3  
ON = 6-x = 6-1 = 5

The triangle MON is a 3,4,5 right triangle.

Edwin

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