Question

In the given figure, points G, D, E, F are points on a circle with centre C. If $\angle ECF = 70^\circ$ and $m(\text{arc } DGF) = 200^\circ$, find: 
(i) $m(\text{arc } DE)$ 
(ii) $m(\text{arc } DEF)$ 

Hint:

In a circle, the measure of a major arc and minor arc always add up to $360^\circ$. Use this property to find unknown arcs.

Answer 1

Step 1: Recall the relationship between central angle and its intercepted arc.
The measure of an arc is equal to the measure of its corresponding central angle.
Step 2: For arc DE.
Given that $\angle ECF = 70^\circ$, \[ m(\text{arc } EF) = 70^\circ \] Also, the total measure of the circle is $360^\circ$.
Given $m(\text{arc } DGF) = 200^\circ$, so the remaining part of the circle (arc DEF) is: \[ m(\text{arc } DEF) = 360^\circ - 200^\circ = 160^\circ \] Now, \[ m(\text{arc } DEF) = m(\text{arc } DE) + m(\text{arc } EF) \] \[ 160^\circ = m(\text{arc } DE) + 70^\circ \] \[ m(\text{arc } DE) = 90^\circ \]
Step 3: Final values.
(i) $m(\text{arc } DE) = 90^\circ$
(ii) $m(\text{arc } DEF) = 160^\circ$
Correct Answers:
(i) $m(\text{arc } DE) = 90^\circ$
(ii) $m(\text{arc } DEF) = 160^\circ$