A Nice geometry math question. | Find x

We are asked to find x in this diagram. Welcome back to my channel. First of all, let's consider triangle ABC. Okay, now triangle ABC is a right angle triangle. So using Pythagorean theorem, using Pythagorean theorem, we are going to have that AB squared AB squared plus BC squared BC squared should be equal to AC squared. AC squared. Okay. All right. Now what is our AB? AB is eight. So we have eight squared plus BC squared, that is six squared, is equal to AC squared. We don't know yet. Okay, so AC squared. Now, when you look at this, you are going to notice something. We have eight squared will give us 64, six squared will give us 36, is equal to AC squared. 64 plus 36 is 100, is equal to AC squared. Now let's take square root of both sides. Let's take square root of both sides. So if we do that, we'll have square root of 100 is equal to square root of AC squared. Now, square root of 100 is plus or minus 10, is equal to AC. But remember that AC is a length. So the length cannot be negative. Therefore, our AC is 10. So let's write 10 here. Please subscribe to the channel and turn on your notification bell if this is the first time you are seeing us. Okay. All right. Now, remember, there is a triangle theorem that says that the sum of the two opposite angles, okay, this is what I mean, is equal to the external angle. Now, if this is X, this is A, right? And this is B. Now, A plus B may be equal to X. Okay. This is equal to the sum of the two opposite interior angles. So now let us relate that to what we have here. Let us consider triangle ABC. A triangle ABC. So this is what we have. This is A. This is B. And this is C. So this is a right angle. And we have alpha here. All right. Okay. So now let me introduce what we have here. We have this. So it means that the whole of this should be alpha plus 90. Alpha plus 90 degrees. But already we have 90 degrees here. Here is already 90 degrees. So which simply means that the remaining portion, which should be this side, may also be what should be alpha. So that we have alpha plus 90 for the whole of this place. So if this place is alpha, then this place should be alpha as well. All right. Now having done that, I want us to take a look at this variable. These two triangles. You can say that triangle BAC is equal to angle DCE. Okay. You have put alpha. Also triangle DEC or CED. Okay. Let's start with this triangle. Angle ABC is equal to angle DEC or CED. So you notice that they share two triangles. Two angles are equal already. That means these two triangles or these two angles are similar. Right. And if they are similar, then it simply means that the ratio of their corresponding sides are equal. If they are similar, the ratio of their corresponding sides are equal. Okay. So now let's go ahead and solve. So if I take AB divided by the side that is holding our alpha and our 90 degrees is this. Alpha and 90 degrees. So it should be over CE. Right. Is equal to. Now let's take AC over. Now AC is holding our alpha and the unknown triangle. Unknown angle. So alpha and unknown. So let us now bring everything. Right. Let's substitute. Now we are going to have AB is 8 over CE is the side that we are looking for. X is equal to AC is 10. We are going to solve this using two methods. So this is the first method. All right. And we have CED which is 18. So cross multiplying the second method is actually very easy. Okay. So cross multiplying we have 10X is equal to 8 times 18. So dividing both sides by 10, we are going to have 10X over 10 is equal to 8 multiplied by 18 over 10. Then we divide 10 and we have X is equal to. Now here 2 we divide this to give us 4. 2 we divide this to give us 5. Now 4 times 18. 4 times 8 is 32. Right. 2 times 3. 4 times 1 is 4 plus 3 is 7. So we have 72 divided by 5 units. We can actually change this to decimal. Okay. And still get our answer. So this is our answer in fraction. The answer to CE. Now the second method that is very easy. Like I said, hope you have subscribed to this channel. Please subscribe. Turn on your notification bell as well. All right. Now the second method. For the second method, we already have some facts stated. So we already know that AC is 10. And we already know that this place is X. Okay. Now in triangle ABC, in triangle ABC, if I say cos alpha, remember that cos is adjacent over hypotenuse. Now the adjacent is 8 and the hypotenuse is 10. All right. So we have 8 over 10. If we reduce this to its lowest term, we will have 4 over 5. Okay. So we are going to keep this Now in triangle DCE or CDE. In triangle CDE, cos alpha is equal to adjacent over hypotenuse. That is X over 18. Okay. Now since cos alpha is this in equation 1 and is also this in equation 2. So it simply means that equation 1 is equal to equation 2. Right. So we can write that 4 over 5 is equal to X over 18. Now if we cross multiply, we have 5X is equal to 4 times 18. Okay. So divide by 5. Divide by 5. We have X is equal to. Now 4 times 18 gives us 72. Right. So 72 over 5. Just like we got using the other method. Thank you for watching.