If the centroid of the triangle ABC with vertices A(5,-4), B(7,8) and C(a,b) is G(7,6), then (b-a) is
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To quickly find a missing vertex when the centroid and two other vertices are known, use the linear shortcut:
\[
\text{Missing Coordinate} = 3 \times (\text{Centroid Coordinate}) - (\text{Sum of other two coordinates})
\]
• a = 3(7) - (5 + 7) = 21 - 12 = 9
• b = 3(6) - (-4 + 8) = 18 - 4 = 14
This direct formula eliminates fractions immediately!
Step 1: Recall the centroid formula. The centroid of a triangle is the average of its three vertices. So $X = \frac{x_1+x_2+x_3}{3}$ and $Y = \frac{y_1+y_2+y_3}{3}$.
Step 2: List the known points. Vertices are $A(5,-4)$, $B(7,8)$, $C(a,b)$, and the centroid is $G(7,6)$.
Step 3: Solve for $a$ using the $x$ values. \[ \frac{5 + 7 + a}{3} = 7 \;\Rightarrow\; 12 + a = 21 \;\Rightarrow\; a = 9 \]
Step 4: Solve for $b$ using the $y$ values. \[ \frac{-4 + 8 + b}{3} = 6 \;\Rightarrow\; 4 + b = 18 \;\Rightarrow\; b = 14 \]
Step 5: Form the required difference. The question asks for $b - a$, and we now have $b = 14$ and $a = 9$.
Step 6: Subtract. \[ b - a = 14 - 9 = 5 \] So the answer is \[ \boxed{5} \]