Can YOU Find The Missing Length? – Geometry Puzzle

Hello my lovelies, it's Susanna and today I want to show you how to solve this problem. Given is a square with two line segments inside that intersect at a right angle. We can see our square here and inside we have this line segment and this line segment and they intersect at this right angle here. We are given the length of this part of this part and of this part and the question is can you find the missing length? This length here is what we're looking for. I call this part x and how can we find this length? Well first of all we keep in mind that we work with a square here so we know that this length here is the same as this length here. We are given this length here and we also have this length down here again so maybe we just draw a line down to here perpendicular so that we can say that this part down here is of length four as well. And now we divided this side here into three parts. We know the length of this one, we know the length of this one, well we don't know the length of this one here but I just call it a. And we could do the same thing with the seven here so go from here perpendicular to the side so that we know that this here is of length seven and then we also divided this side here into three parts where we know this length, well we don't know this length here and we also don't know this length so I call this length here b. But we created triangles. Can you see them? We have one triangle here and another triangle here and if we take a look at them they look pretty similar. It looks like this orange triangle if we just turn it then we get this green triangle. And how would this help us if we knew that these two triangles were actually congruent to each other? Well then we would know that the lengths of the sides in both triangles were the same so we would know that this side here has the same length as this side here and this would simplify things for us. So let's try and prove that these two triangles are actually congruent to each other. How can we do this? We have to take a look at the sides and the angles of our two triangles and compare them. So if we start with this side here in our green triangle then this has the same length as the side of the square and in our orange triangle we also have such a side here. This side has the length of the side of the square. So in both triangles we have a side of the same length which is great because now we can take a look at the angles on this one particular side. So here on our side here to the right we have a right angle. Okay we have a right angle here. Do we also have this right angle in our orange triangle on this side? Yes it is there so which is great. So maybe we can use the congruence theorem ASA which says if you have two equal angles in your triangles and the included side is also of the same length then your triangles are automatically congruent to each other. Okay we have one angle of the same size we have the side of the same length. What about this angle on the other side then? So here this angle here if I call it alpha can we find alpha in our orange triangle here then as well? So can we prove that this angle here has the same size? Well let's try it. I mean we have this green triangle where we have alpha and 90 degrees angle and I call this third angle beta then. So we know now that every time we are in a triangle where we have a right angle already in there and one angle is called alpha then the other angle has to be equal to beta because we have more triangles here. So for example here this triangle we have alpha in here and this angle because it lies across this right angle is then automatically also a right angle. So in this small triangle we have a right angle we have alpha and we just said then the third angle has to be beta. So we have beta down here. Okay but there are more triangles this one for example here. Here we have beta in here already. What about this angle here? This is a right angle because we went down perpendicular here and we went perpendicular here to the side so we have a right angle here. In this triangle then we have a right angle we have beta well then the third angle has to be equal to alpha. So we just showed that our two triangles are congruent to each other and now we are allowed to use that this side here that we called A has the same length as this side here that we called B. So I don't have to call it B I can just call it A. Okay this is what we have so far. We are still interested in our X in the end and X is part of this side of the square which has the same length as this side of the square. So now we can compare these two lengths. So if I add these here 4 plus A plus 6 then this has to be the same as my X plus A plus 7. Let's try and solve this equation for our X. We have A in here A in here if we subtract A, A will not longer be with us. So we have 4 here this cancels out we have the plus 6 here on the other side we have our X this cancels out and the plus 7. 4 plus 6 equals 10 so to solve for X we subtract 7 on both sides of the equation 10 minus 7 equals 3 and this is our X and we solved our problem. I'm curious how you solved this problem so please let me know in the comments. I wish you a wonderful day and I hope to see you in one of my next videos. Take care!