Question

point m is the midpoint of fg. can the value of variable a be determined? explain.
a. 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
b. 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
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

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here, if $F(b + 1,a + 2)$ and $G(2a,3b+3)$ and $M(3,5)$, then for the x - coordinate: $\frac{(b + 1)+2a}{2}=3$, which simplifies to $b + 1+2a=6$ or $b=5 - 2a$. For the y - coordinate: $\frac{(a + 2)+(3b + 3)}{2}=5$, which simplifies to $a+2 + 3b+3 = 10$ or $a+3b=5$.

Step2: Substitute $b$ into the second equation

Substitute $b = 5 - 2a$ into $a + 3b=5$. We get $a+3(5 - 2a)=5$. Expand: $a + 15-6a=5$. Combine like terms: $- 5a=5 - 15=-10$. Solve for $a$: $a = 2$. But we had to set up and solve a system of equations in two variables ($a$ and $b$).

Answer:

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