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Point M is the midpoint of FG overbar. Can the value of variable a be​ determined? Explain.

Upper F left parenthesis b plus 1 comma a plus 2 right parenthesisUpper M left parenthesis 3 comma 5 right parenthesisUpper G left parenthesis 2 a comma 3 b plus 3 right parenthesis

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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
To determine the value of​ a, set up and solve a system of equations in two variables. The value of a is
  
enter your response here.
B.
To determine the value of​ a, set the​ x- and​ y-coordinates of G and F equal to each other and solve for a. The value of a is
  
enter your response here.
C.
To determine the value of​ a, divide each coordinate of M by 2. Then add each quotient to the​ x- and​ y-coordinates of M. The value of a is
  
enter your response here.
D.
Not enough information is given for the value of a to be determined.

1 answer

To determine the value of variable \( a \) using the information given that point \( M \) is the midpoint of \( \overline{FG} \), we can use the properties of midpoints in coordinate geometry.

The coordinates of point \( F \) are \( (b + 1, a + 2) \), the coordinates of point \( G \) are \( (2a, 3b + 3) \), and the coordinates of the midpoint \( M \) are \( (3, 5) \).

The midpoint \( M \) of line segment \( \overline{FG} \) can be calculated using the following formulas for the \( x \)-coordinate and the \( y \)-coordinate:

\[ M_x = \frac{x_F + x_G}{2} \quad \text{and} \quad M_y = \frac{y_F + y_G}{2} \]

Substituting the coordinates of points \( F \) and \( G \), we have:

\[ M_x = \frac{(b + 1) + (2a)}{2} \quad \text{and} \quad M_y = \frac{(a + 2) + (3b + 3)}{2} \]

Setting \( M_x \) and \( M_y \) equal to the coordinates of midpoint \( M \) (which are \( 3 \) and \( 5 \)), we get the following system of equations:

  1. \(\frac{(b + 1) + (2a)}{2} = 3\)
  2. \(\frac{(a + 2) + (3b + 3)}{2} = 5\)

Simplifying the equations:

  1. For the first equation: \[ (b + 1) + 2a = 6 \implies b + 2a = 5 \quad \text{(Equation 1)} \]

  2. For the second equation: \[ (a + 2) + (3b + 3) = 10 \implies a + 3b + 5 = 10 \implies a + 3b = 5 \quad \text{(Equation 2)} \]

Now we have the following system of equations:

  1. \( b + 2a = 5 \)
  2. \( a + 3b = 5 \)

Solving the system of equations:

You can solve this system using substitution or elimination. However, the equations contain two variables \( a \) and \( b \), which means that we need both variables to determine a unique solution for \( a \).

Conclusion:

Since we cannot uniquely determine \( a \) without knowing \( b \), the correct choice is:

D. Not enough information is given for the value of a to be determined.