Given: AB = 28; M lies on the line AB
Prove: M is a midpoint.
Statements
Reasons
1) M lies on AB
AB = 28
1)
2) AM + MB = AB
2)
3) 2x + 2 + 4x - 10 = 28
3) Substitution Property
4) 6x - 8 = 28
4)
5) 6x = 36
5)
6) x = 6
6)
7) AM = 2(6) + 2 and
MB = 4(6) - 10
7) Substitution Property
8) AM = 14 and MB = 14
8) Simplifying
9) M is a midpoint
9)
10) AM = MB
10) Definition of a midpoint
To prove that M is a midpoint, we need to show that AM + MB = AB.
Given that AB = 28 (from statement 1), we can substitute this value into the equation.
We know that AM + MB = AB, so we substitute AM = 2x + 2 and MB = 4x - 10 (from statement 7).
This gives us the equation:
(2x + 2) + (4x - 10) = 28
Simplifying this equation, we get:
6x - 8 = 28
To solve for x, we add 8 to both sides of the equation:
6x = 36
Finally, we divide both sides by 6 to isolate x:
x = 6
Now that we have the value of x, we can substitute it back into the equations for AM and MB:
AM = 2(6) + 2 = 14
MB = 4(6) - 10 = 14
From statement 8, we can see that AM and MB are both equal to 14. This means that both segments are equal in length, which proves that M is a midpoint (statement 9).