Question

A farmer has a circular garden as shown in the picture above . He has a different type of tree ,plants and flower plants in his garden. In the garden, there are two mango trees A and B at a distance of AB=10m.Similarly has two Ashok trees at the same distance of 10m as shown at C and D AB subtends ∠AOB=1200 at the center O, The perpendicular distance of AC from center is 5m the radius of the circle is 13m.


What is the distance between mango tree A and Ashok tree C ?

What is the value of ∠OAB ?


What is the value of ∠OCD ?


What is the value of ∠ODC ?


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Answer 1

Find the given angles based on the figure.

Explanation:

• As we know that a perpendicular line drawn from the center of a circle to a chord of the circle meeting it at a point divided the said chord into two equal parts.
• Hence, let there be a perpendicular drawn from O to the chord AB meeting at point M. Hence, $AM=MB$
• Since its given $AB=10\ m\ \ \ \ \ ->AM=MB=5\ m$  
• Now, in the triangle OAM angle M is a right angle hence, OA is the hypotenuse and AM is the base.
• We get, $sin(\angle OAB)= \frac{AM}{OA}=\frac{5}{13}\\ ->\angle OAB= sin^{-1}\frac{5}{13}= 67$  
• Hence the measure of angle OAB is $67$°
• Similarly we get the measure of angle OCD is $67$°
• in triangles AOB and COD pair of angles OAB, OBA and OCD , ODC are equals.
• hence, the measure of angle ODC is $67$°