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Circles — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsCircles1 Mark
MCQ
In the given figure, $ABCD$ is a quadrilateral inscribed in circle with centre $O$. $CD$ is produced to $E$. If $\angle\text{ADE}=95^\circ$ and $\angle\text{OBA}=30^\circ,$ then $\angle\text{OAC}$ is equal to:
  • A
    $10^\circ$
  • B
    $20^\circ$
  • C
    $15^\circ$
  • ✓
    $5^\circ$
✓

Answer

Correct option: D.
$5^\circ$

Here, $\angle\text{ADC}$ and $\angle\text{ADE}$ are supplementary to each other,
So, $\angle\text{ADC}=180-95=85^\circ$
Also, $ABCD$ is also a cyclic quadrilateral so, $\angle\text{ADC}$ and $\angle\text{ABC}$ are supplementary
So, $\angle\text{ABC}=\angle\text{ADC}=180^\circ$
$\angle\text{ABC}=180-95=85^\circ$
angle subtended at centre is double the angle suntended at circumference.
So, $\angle\text{AOC}=170^\circ$
Now, in triangle $AOC, OA = OC($Radii$)$ so, $\angle\text{OAC}=\text{OCA}$
$\angle\text{OCA}+\angle\text{OAC}+\angle\text{AOC}=180^\circ$
$2\angle\text{OAC}=180-170=10^\circ$
$\angle\text{OAC}=5^\circ$

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