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Question

Part 1
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

x y graph
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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.

Answer 1

To determine if the value of variable \( a \) can be found using the given midpoint \( M \) of line segment \( FG \), we begin by using the midpoint formula. The midpoint \( M \) of a segment with endpoints \( F(x_1, y_1) \) and \( G(x_2, y_2) \) is calculated as:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Given:

• \( F(b + 1, a + 2) \)
• \( M(3, 5) \)
• \( G(2a, 3b + 3) \)

From the midpoint \( M \) coordinates, we can set up two equations based on the \( x \)-coordinates and \( y \)-coordinates.

1. For the \( x \)-coordinates: \[ \frac{(b + 1) + (2a)}{2} = 3 \]

2. For the \( y \)-coordinates: \[ \frac{(a + 2) + (3b + 3)}{2} = 5 \]

Now, let's solve these equations:

For the \( x \)-coordinates:

Multiply both sides by \( 2 \): \[ (b + 1) + 2a = 6 \quad \Rightarrow \quad b + 2a = 5 \quad \text{(Equation 1)} \]

For the \( y \)-coordinates:

Multiply both sides by \( 2 \): \[ (a + 2) + (3b + 3) = 10 \quad \Rightarrow \quad a + 3b + 5 = 10 \quad \Rightarrow \quad a + 3b = 5 \quad \text{(Equation 2)} \]

We now have a system of equations:

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

To solve this system for \( a \) and \( b \), we can express \( b \) from Equation 1: \[ b = 5 - 2a \]

Now substitute \( b \) into Equation 2: \[ a + 3(5 - 2a) = 5 \ a + 15 - 6a = 5 \ -5a + 15 = 5 \ -5a = 5 - 15 \ -5a = -10 \ a = 2 \]

Since we found \( a = 2 \), we have determined a specific value for \( a \).

Conclusion:

So the correct choice is:

A. To determine the value of \( a \), set up and solve a system of equations in two variables. The value of \( a \) is 2.