00:01
In this problem, a circle is given whose radius is given as 7 cm and a cord whose distance from the center of the circle is also given 4 cm.
00:13
And in the problem, they are asking two things.
00:15
First is, what is the angle, the smaller angle subtended by the cord on the center of the circle? second thing is the length of the smaller arc made by the cord.
00:28
So let me draw the circle first.
00:31
Let me draw the figure to make it more clear circle like this and let's call this as the center and let's say the chord is this yeah okay now i'm going to call this card as a b yes so this is the radius of the circle and center i'll be calling it as o so ab is your card if a b is the cord so from a and b.
01:09
Okay.
01:10
Now, one more information given is the distance from center of the circle to the chord.
01:17
Okay.
01:17
Let me call this point as c.
01:19
This is going to be 4 centimeter.
01:24
Oc is going to be 4 centimeter.
01:28
Oa and ob is going to be 7 centimeter because the radius is given as 7 centimeter.
01:33
One more important point here is oc is always perpendicular to ab.
01:37
That is how we measure distance from a point to a line okay from center to cord it should be a perpendicular light to the cord that is called the distance okay so this is perpendicular now let me write down what is given first radius radius i'll call it a small r is equal to seven centimeter yeah which is also equal to oa which is also equal to o b clearly can see o to a is also a radius o to b is also a radius and one more thing is given o to c is the distance from center of the circle to the chord which is given as 4 centimeters okay so these are the two informations given now first of all let me let us look at triangle oca this triangle here this triangle left side triangle triangle oca if you look at that we can see it is a right angle triangle why because angle oca this angle is 90 degree so it's a right angle triangle now this part let me call this as this angle as theta 1 and let me call this angle as theta 2 okay so two angles are the theta 1 and theta 2 theta 1 is for triangle oca okay so angle angle a o c i'm going it going to call it as theta 1 angle b oc i'm going to call it as theta 2.
03:13
Now, let us get back to a triangle oca.
03:17
So in triangle oca, what are we going to find? so, theta 1, actually, cost theta 1, okay? cos theta 1 is basically adjacent by hypotenuse.
03:29
We know cause of any angle is adjacent side to that angle and the hypotenuse.
03:35
Okay.
03:36
So in triangle oca to theta 1, which is adjacent to.
03:40
Side that is side 4 okay or side oc divided by hypothesis oa so oc we already know it's 4 oa is 7 okay so this will be the cost theta value which is going to be 0 .5714 okay now so from there you can find theta 1 value is going to be 55 .15 degree this is theta 1 value now same way in triangle o cb also you can find theta 2 again here cost theta 2 is going to be oc divided by ob which is also same as 4 divided by 7 same thing see cost theta 2 here theta 2 is here so adjacent is 4 ipodness is 7 4 by 7 yeah so same value of angle you are going to get theta 2 is equal to 55 .15 degree so, but theta 1, theta 2 is not the question they have asked us.
04:47
They have asked the total angle, okay? the total angle subtended by this arc ab, okay? arc ab is i'll put it as right...