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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:

$$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

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 $2$: $$(b+1)+2a=6\Rightarrow b+2a=5\text{(Equation 1)}$$

For the $y$-coordinates:

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

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: $$a+3(5-2a)=5a+15-6a=5-5a+15=5-5a=5-15-5a=-10a=2$$

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.