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Circles
CBSE 2026 Board Exam Set 1 (Code 30/7/1)
CBSE_BOARD_PYQ_2026_30_7_1
Grade 10
Question:
[Section B]
In the given figure, $AC$ is a chord of a circle with center $O$ and $AB$ is a tangent at $A$. If $\angle AOC$ is a straight line segment passing through $O$ and $OA = OC$, prove that $\angle BAC = \angle DCA$.
In the given figure, $AC$ is a chord of a circle with center $O$ and $AB$ is a tangent at $A$. If $\angle AOC$ is a straight line segment passing through $O$ and $OA = OC$, prove that $\angle BAC = \angle DCA$.
Step-by-Step Solution
Key Concept: Radius is perpendicular to tangent at point of contact.
Since $OA = OC$, in $\triangle OAC$, $\angle OAC = \angle OCA$. Eq (1). [0.5 Mark]
Tangent is perpendicular to radius: $\angle OAB = 90^\circ$. [0.5 Mark]
$\angle BAC = 90^\circ - \angle OAC = 90^\circ - \angle OCA = \angle DCA$. Hence proved. [1.0 Mark]
Tangent is perpendicular to radius: $\angle OAB = 90^\circ$. [0.5 Mark]
$\angle BAC = 90^\circ - \angle OAC = 90^\circ - \angle OCA = \angle DCA$. Hence proved. [1.0 Mark]
Correct Answer: Proof via angle properties of radius and tangent.
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