Straight Lines
Straight Lines
nta_abhyas_2025
Grade 11
Question:
Find the coordinates of vertex $C$ if the centroid is at $G = (1, \frac{5}{3})$, with vertices $A(\frac{11}{4}, \frac{4}{3})$ and $B(-\frac{11}{3}, \frac{1}{3})$ given.
Step-by-Step Solution
Key Concept: The centroid of a triangle is the average of its three vertices' coordinates.
The centroid of a triangle divides each median in the ratio $2:1$. Using the centroid formula $G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$, we have $1 = \frac{\frac{11}{4} + (-\frac{11}{3}) + x_C}{3}$ and $\frac{5}{3} = \frac{\frac{4}{3} + \frac{1}{3} + y_C}{3}$. Solving these equations: from the first equation, $3 = \frac{11}{4} - \frac{11}{3} + x_C$, which gives $x_C = 1$. From the second equation, $5 = \frac{5}{3} + y_C$, which gives $y_C = -\frac{10}{3}$. Thus $C = (1, -\frac{10}{3})$.
Correct Answer: 4