Question

point n is the midpoint of $overline{gb}$. $gn = 7r - 3$, $bn = 5r + 9$. what is $moverline{gb}$?

Explanation:

Step1: Use mid - point property

Since N is the mid - point of $\overline{GB}$, then $GN = BN$. So, we set up the equation $7r−3 = 5r + 9$.

Step2: Solve the equation for r

Subtract $5r$ from both sides: $7r-5r−3=5r - 5r+9$, which simplifies to $2r−3 = 9$. Then add 3 to both sides: $2r−3 + 3=9 + 3$, giving $2r=12$. Divide both sides by 2: $r=\frac{12}{2}=6$.

Step3: Find the length of GN or BN

Substitute $r = 6$ into the expression for $GN$ (we could also use $BN$). $GN=7r−3=7\times6−3=42 - 3=39$.

Step4: Calculate the length of GB

Since $GB=GN + BN$ and $GN = BN$, then $GB = 2\times GN$. So, $GB=2\times39 = 78$.

Answer:

78