Coordinate Geometry
RD Sharma
CBSE
Grade 10
Question:
If $A(-2, -1), B(a, 0), C(4, b)$ and $D(1, 2)$ are the vertices of a parallelogram $ABCD$, find the values of $a$ and $b$.
Step-by-Step Solution
Key Concept: Midpoint of $AC = \left(\dfrac{-2+4}{2}, \dfrac{-1+b}{2}\right) = \left(1, \dfrac{b-1}{2}\right)$. Midpoint of $BD = \left(\dfrac{a+1}{2}, \dfrac{0+2}{2}\right) = \left(\dfrac{a+1}{2}, 1\right)$.<br>$\dfrac{a+1}{2} = 1 \Rightarrow a = 1$ and $\dfrac{b-1}{2} = 1 \Rightarrow b = 3$.
Midpoint of $AC = \left(1, \dfrac{b-1}{2}\right)$ and Midpoint of $BD = \left(\dfrac{a+1}{2}, 1\right)$. [0.5 Mark]
$\dfrac{a+1}{2} = 1 \Rightarrow a = 1$. [0.5 Mark]
$\dfrac{b-1}{2} = 1 \Rightarrow b = 3$. Values: $a = 1, b = 3$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Equating midpoints of diagonals: 0.5 Mark
Solving $a = 1$: 0.5 Mark
Solving $b = 3$: 1.0 Mark
Correct Answer:
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