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Coordinate Geometry
NCERT Exemplar
CBSE
Grade 10

Question:

If $A(2, 2), B(-4, -4)$ and $C(5, -8)$ are the vertices of $\Delta ABC$, find the length of the median through vertex $C$. Also find the coordinates of the centroid of $\Delta ABC$.

Step-by-Step Solution

Key Concept: Median through $C$ joins $C$ to the midpoint $D$ of $AB$. Centroid $G = \left(\dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3}\right)$.
Midpoint $D$ of side $AB = \left(\dfrac{2 + (-4)}{2}, \dfrac{2 + (-4)}{2}\right) = \left(\dfrac{-2}{2}, \dfrac{-2}{2}\right) = (-1, -1)$. [1.5 Marks]
Length of median $CD = \sqrt{(5 - (-1))^2 + (-8 - (-1))^2} = \sqrt{6^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85}$ units. [1.5 Marks]
Centroid $G = \left(\dfrac{2 + (-4) + 5}{3}, \dfrac{2 + (-4) + (-8)}{3}\right) = \left(\dfrac{3}{3}, \dfrac{-10}{3}\right) = \left(1, -\dfrac{10}{3}\right)$. [2.0 Marks]

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🎯 Official CBSE Marking Scheme:
Finding midpoint $D(-1, -1)$ of $AB$: 1.5 Marks
Calculating median length $CD = \sqrt{85}$ units: 1.5 Marks
Calculating centroid coordinates $G(1, -10/3)$: 2.0 Marks

Correct Answer:
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