Question 601836: Hello,
I have difficulty with a question in a test study guide I am working on. It has a figure of a circle with 2 segments parallel to each other. The problem is written below:
In the figure above, lines AE and BD are parallel, and line AD is the diameter of the circle. If the length of arc BCD is equal to 4π, then what is the area of the circle?
Lines AE and BD are the chords of each segment and between B and D is the arc C. Also there is a line of the diameter of the circle connecting A and D. The angle of EAD is 30 degrees. That is all the information given.
I am more interested in the correct formula to use than the answer. So please just show me the method to get the area of the circle using this information. Thank you for your help.
Answer by w_parminder(53)   (Show Source):
You can put this solution on YOUR website! Hello dear,
In order to solve this problem, you have to first find the angle made by the arc BCD at the center of the circle,
Then use the formula to find the length of the arc
i.e. Length of arc = (π*r*θ)/180
where 'θ' is the angle made by the arc at the center,
'r' is the radius of the circle.
Using this formula, find the radius and then the area of the circle.
Area of circle = (π*r^2)
THE SOLUTION
AE is parallel to BD,
So angle EAD = angle ADB (because these are interior alternate angles)
So angle ADB = 30 degrees
Let 'O' be the center of the circle, so it must lie on the diameter AD.
Join OB
Now in triangle OBD,
OB = OD [radii of the circle]
As you know, the angles opposite to the equal sides of a triangle are equal,
So, angle ODB = angle OBD
But angle ODB (OR ADB) = 30 degrees
So angle OBD = 30 degrees
Now using the angle sum property of a triangle, we will find the measure of angle BOD which is 120 degrees
Now Length of arc = (π*r*θ)/180
Solving this, we get r = 6 units
Area of circle = (π*r^2)
= 3.14*6*6
= 113.14 square units
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