Question

point m is the mid - point of fg. can the value of variable a be determined? explain. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. 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 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 c. to determine the value of a, set up and solve a system of equations in two variables. the value of a is d. not enough information is given for the value of a to be determined.

Explanation:

Step1: Recall mid - point formula

If $M(x_m,y_m)$ is the mid - point of the line segment joining $F(x_f,y_f)$ and $G(x_g,y_g)$, then $x_m=\frac{x_f + x_g}{2}$ and $y_m=\frac{y_f + y_g}{2}$. Here, $x_m = 3$, $y_m=5$, $x_f=b + 1$, $y_f=a + 2$, $x_g=2a$, $y_g=3b+3$. We get two equations: $3=\frac{b + 1+2a}{2}$ and $5=\frac{a + 2+3b+3}{2}$. These are two equations with two variables $a$ and $b$. To find $a$, we need to set up and solve this system of equations.

Step2: Analyze answer choices

Choice A is incorrect as the method described is wrong. Choice B is incorrect because we cannot just set the coordinates of $G$ and $F$ equal. We use the mid - point formula to set up a system of equations.

Answer:

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