In the given figure, O is the centre of the circle. Where, central angle AOC is equal 140 degrees. a. Write…

Question

In the given figure, O is the centre of the circle. Where, central angle AOC is equal 140 degrees. a. Write the relation between the inscribed angle ADC and the arc ABC. b. Find the value of angle ADC from the given figure. c. Experimentally verify that the opposite angles ABC and ADC of cyclic quadrilateral ABCD are supplementary.

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12. a) परिकोष कोण ADC र चाप ABC बिचको सम्बन्ध लेख्नुहोस्। (Write the relation between the inscribed angle ADC and the arc ABC.)

State the relationship between an inscribed angle and its intercepted arc.
An inscribed angle is half the measure of its intercepted arc.
$\angle ADC = \frac{1}{2} \text{arc(ABC)}$
The relation is $\boxed{\angle ADC = \frac{1}{2} \text{arc(ABC)}}$ .

12. b) दिइएको चित्रबाट $\angle ADC$ को मान पत्ता लगाउनुहोस्। (Find the value of $\angle ADC$ from the given figure.)
Identify the given central angle.
The central angle $\angle AOC = 140^\circ$ . This angle intercepts the minor arc ADC.
Find the measure of the major arc ABC.
The sum of angles around the center is $360^\circ$ . The major arc ABC is intercepted by the reflex angle $\angle AOC$ .
$\text{arc(ABC)} = \text{reflex } \angle AOC = 360^\circ - \angle AOC$ $\text{arc(ABC)} = 360^\circ - 140^\circ$ $\text{arc(ABC)} = 220^\circ$
Use the relation from part a) to find $\angle ADC$ .
The inscribed angle $\angle ADC$ intercepts arc ABC.
$\angle ADC = \frac{1}{2} \text{arc(ABC)}$ $\angle ADC = \frac{1}{2} \times 220^\circ$ $\angle ADC = 110^\circ$
The value of $\angle ADC$ is $\boxed{110^\circ}$ .

12. c) चक्रीय चतुर्भुज ABCD का सम्मुख कोणहरू ABC र ADC परिपूरक हुन्छन् भनी प्रयोगात्मक विधिबाट प्रमाणित गर्नुहोस्। (कम्तीमा 3 से.मि. अर्धव्यास भएका दुईओटा वृत्तहरू आवश्यक छन्।) (Experimentally verify that the opposite angles ABC and ADC of cyclic quadrilateral ABCD are supplementary. (Two circles with at least 3 cm radii are necessary.))

To experimentally verify that the opposite angles of a cyclic quadrilateral are supplementary, follow these steps:
Draw two circles with a radius of at least 3 cm.
Label the center of each circle as O.
For each circle, mark four points A, B, C, and D on the circumference in order to form a cyclic quadrilateral ABCD.
Connect these points to form the quadrilateral.
Using a protractor, carefully measure the interior angles $\angle ABC$ and $\angle ADC$ for each quadrilateral.
Add the measured values of $\angle ABC$ and $\angle ADC$ .
You should find that for both quadrilaterals, $\angle ABC + \angle ADC \approx 180^\circ$ . Due to measurement errors, the sum might not be exactly $180^\circ$ , but it should be very close.
Conclude that the opposite angles of a cyclic quadrilateral are supplementary.

Step-by-step solution

State the relationship between an inscribed angle and its intercepted arc.

Identify the given central angle.

Find the measure of the major arc ABC.

Use the relation from part a) to find $\angle ADC$ .

Draw two circles with a radius of at least 3 cm.

For each circle, mark four points A, B, C, and D on the circumference in order to form a cyclic quadrilateral ABCD.

Using a protractor, carefully measure the interior angles $\angle ABC$ and $\angle ADC$ for each quadrilateral.

Add the measured values of $\angle ABC$ and $\angle ADC$ .

Conclude that the opposite angles of a cyclic quadrilateral are supplementary.

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