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Triangles
NCERT Exemplar
CBSE
Grade 10
Question:
In a trapezium $ABCD$ with $AB \parallel DC$, the diagonals $AC$ and $BD$ intersect at $O$. If $AB = 3 CD$, then the ratio $\dfrac{AO}{CO}$ is:
(a) $1 : 3$
(b) $3 : 1$
(c) $1 : 9$
(d) $9 : 1$
(a) $1 : 3$
(b) $3 : 1$
(c) $1 : 9$
(d) $9 : 1$
Step-by-Step Solution
Key Concept: $\Delta AOB \sim \Delta COD$ by AA similarity because alternate interior angles are equal.
In $\Delta AOB$ and $\Delta COD$: $\angle OAB = \angle OCD$ (alt int angles) and $\angle AOB = \angle COD$ (vert opp angles). [0.5 Mark]
$\Delta AOB \sim \Delta COD \Rightarrow \dfrac{AO}{CO} = \dfrac{AB}{CD} = \dfrac{3 CD}{CD} = 3 : 1$. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Proving similarity of $\Delta AOB$ and $\Delta COD$: 0.5 Mark
Evaluating ratio $AO/CO = 3:1$: 0.5 Mark
$\Delta AOB \sim \Delta COD \Rightarrow \dfrac{AO}{CO} = \dfrac{AB}{CD} = \dfrac{3 CD}{CD} = 3 : 1$. [0.5 Mark]
---
🎯 Official CBSE Marking Scheme:
Proving similarity of $\Delta AOB$ and $\Delta COD$: 0.5 Mark
Evaluating ratio $AO/CO = 3:1$: 0.5 Mark
Correct Answer: $3 : 1$
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