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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.

1 answer

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(x1,y1) and G(x2,y2) is calculated as:

M=(x1+x22,y1+y22)

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: (b+1)+(2a)2=3

  2. For the y-coordinates: (a+2)+(3b+3)2=5

Now, let's solve these equations:

For the x-coordinates:

Multiply both sides by 2: (b+1)+2a=6b+2a=5(Equation 1)

For the y-coordinates:

Multiply both sides by 2: (a+2)+(3b+3)=10a+3b+5=10a+3b=5(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=52a

Now substitute b into Equation 2: a+3(52a)=5 a+156a=5 5a+15=5 5a=515 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.

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