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)
To prove that M is a midpoint, we can follow these steps:
1) Given that M lies on the line AB and AB = 28.
2) According to the definition of a midpoint, we know that AM + MB = AB.
3) By substituting the values of AM and MB using the variable x, we have 2x + 2 for AM and 4x - 10 for MB.
4) Combining the expressions, we have 2x + 2 + 4x - 10 = 28.
5) Applying the Substitution Property.
6) Simplifying the equation further, we get 6x - 8 = 28.
7) Solving for x, we find that 6x = 36, and therefore x = 6.
8) Substituting x = 6 into the expressions for AM and MB, we have AM = 2(6) + 2 = 14 and MB = 4(6) - 10 = 14.
9) Simplifying both expressions, we see that AM = MB = 14.
10) Since AM = MB, M is equidistant from both A and B, making it the midpoint of AB.
11) Therefore, M is a midpoint.