Think you are good at geometry? Try finding x.

In this video we are expected to find the value of x. Welcome back to my channel. So this video asks please this is not drawn to scale and this length is 7 while this length is 2. Now let's name this so that it will be easier for us to solve this. So we have a b c d e this point and this point is g. So first of all before we start the first thing we need to remember is the two tangent theorem. Now the two tangent theorem states that when two tangents are drawn from same point when two tangents to a circle are drawn from the same point. For example you have this is a circle and you have a point and you draw two tangents to the circle from this same point. This particular theorem states that the segments that is this and this that they are equal. They are equal from their point of contact from the point of contact to the point where they emerge from. So if we have that in mind then let's look out for such points on this diagram. Then it simply means that a b and a g are what you can see that both of them are tangent to this circle. This is a semicircle. So note this is 90 degrees. So that means that if this place is 7 then a g is also 7. So permit me to write this that a b is equal to a g which is equal to 7. Now similarly if you look at this as well you can see that from here to here that is d e and e f both are tangents that come from here to this circle this particular circle. So that means that e d is equal to e f which is equal to 2. So it means that this is 2. If this is the first time you're coming across my channel please click the subscription button and turn on your notification bell so that you don't miss our videos. We upload our videos all the time as often as possible. Tune in at your convenience. You must see something new for you to watch. Give us a thumbs up if you love what we are doing. Thank you very much. So moving forward I want us to drop a perpendicular or a line parallel to line a b and line e d. So let's drop a line parallel to line a b and line e d here. If we do that please all these are not drawn to scale. So it has to be parallel and a perpendicular to this place. So that means if this is 90 this is also 90 degrees. Now let's call this point remember that f g is equal to x which is what we are looking for from here to here okay from this point to this point. Now if we call this point e f g h if we call this point h okay now using the same two tangent theorem we will also notice that c h and g h are tangent to this particular circle right. So we also have to write that c h c h is equal to h g and both of them are equal. So if we say that this place is a that means this place is also a okay. So both of them are equal to a. Now you can also see that from f to g to h and from c to h they are also tangents to this particular circle. So both of them should be equal. So it means that c h is equal to h g which is equal to h f which is equal to a okay. Now we have gotten almost all we want that means that h f is a meaning that our x is what a plus which is 2a. So x is equal to 2a. So I don't need f g anymore. So x is equal to 2a. Now let us note remember that when you have when a transversal crosses parallel lines okay when a transversal crosses parallel lines for example these lines are parallel and a transversal crosses them all the corresponding angles are equal right. So if I pick this angle here it will be equal to this angle here which will be equal to this angle here. The same thing if I pick this angle here it will be equal to this angle here and will also be equal to this angle here. So the parallel lines the transversal creates different angles and one of them is corresponding angle and all corresponding angles are equal. Now if you look at what we have on the board we definitely have three parallel lines line a d line h c and line e d. So we have a d we have h c and we have e d both of all of them are parallel okay. So this particular line crossing them means that all these angles are what equal. So it means that angle a is equal to angle h okay. All right now let us draw a line. So this is what our diagram looks like something like this right. Now if we decide to draw a line that is parallel to b d this is our b d and this is a this side is e okay and this is h c h and c. Remember that this angle is equal to this angle right. This is 90 degrees and this is 90 because they are parallel to each other. Now what this is simply telling us is that if we consider triangle a now let me call this y triangle a y e right. So triangle a y e will be similar to triangle h what point should I call this let's call this point i triangle h i e right. Now let's draw both of them out this particular triangle let's call this triangle a y e and let's call this triangle h i okay. So you can see this is equal to this right. This is also equal to this meaning that e and e they share that in common so they are equal. So these two triangles are equal they are similar sorry and what does this mean remember that the ratio of corresponding sides of similar triangles are equal so meaning that we can find the ratio we can equate their ratios right. Now if we say that from a to b is 7 and from e to d is 2 so it means that here is 2 here is 2. Now if a b is 7 it means that a y is 5 right so we have 5 now it means that a y is 5 this is 5. Now for a e a e let's go here from here to here is 7 from here to here is 2 a and from here to here is 2. I didn't write that right it's 2 because of the two tangent theorem right so that means the whole of a e will be 7 plus 2 a plus 2 which is 9 plus 2 a so here is 9 plus 2 a. Now I'm going to this one from here to i from h to i if the whole of this place is a and from here to here is 2 then from h from here sorry let me not confuse myself from h to i then will be what the whole of this which is a minus 2 so it means that here is a minus this side h i is a minus 2 okay and the other side this side is simply 2 plus a so this is 2 plus a. Now we need to redraw this because I need to redraw it so that it will look fine. Now this is what we have remember that the ratio of the corresponding sides are equal right so that means we are going to have 5 over a minus 2 so 5 divided by a minus 2 is equal to 9 plus 2 a divided by 2 plus a okay 9 plus 2 a divided by 2 plus a so let me now clean this and we continue solving so we're going to cross multiply now so this implies that we have 5 brackets 2 plus a is equal to a minus 2 multiplied by 9 plus 2 a so if we open this bracket we have 10 plus 5 a if you open this bracket we have 8 times 9 is 9 a 8 times 2 a is plus 2 a squared minus 2 times 9 is minus 18 and minus 2 times plus 2 a is minus 4 a so we have 10 plus 5 a is equal to 9 minus 4 a is 5 a plus 2 a squared minus 18 so 5 a and 5 a we go right and then we have 10 let's transfer minus 18 to this okay first of all we have 10 is equal to 2 a squared minus 18 sorry I'm actually in a hurry because I want to go somewhere okay so I'm just doing this video let's just do it quickly and leave so um that is why now this is what we have 10 equals 2 a squared minus 18 so if I take this to the other side I'll have 10 plus 18 is equal to 2 a squared let's divide both sides by 2 so if we do that we're going to have 5 plus what 5 plus 9 which is 14 is equal to a squared okay now that means that our a will be square root of 14 and square root of 14 it cannot be reduced more because factor pairs of 14 are 1 and 14 2 and 7 and that's that so it cannot be reduced further that means remember that remember that x is equal to 2 a right so it means that our final answer for x will be 2 root 14 thank you so much for watching and see you in my next video bye