 I've laid out another problem. This one is going to be proof or a solving problem using triangles and the relationship between midpoint and triangles. Specifically, we're going to be talking about congruent triangles and stuff like this, okay? Now, this is the information they've given you and this is the drawing that accompanies the question. So what we're going to have to do is read the question once, find out what they're asking for so don't forget to read what they want so you can visually see where the information is that they've given you and where the information is missing. And then we're going to go through each point and break it down again. So let's read this. O is the midpoint of AW. O is the midpoint of AW. So while we're doing this, we can actually make ticks and show what the information is, transfer the information from here to the drawing. So O is the midpoint of AW. So this guy must be equal to this guy. O is the midpoint of BQ. BQ. Now, we don't know right now if this length from there to there is equal to there to there. So what we're going to do is use two ticks to say that these two guys are equal because we don't know if these two guys are equal, okay? So O is the midpoint of BQ. Therefore, double tick here, double tick here. This guy is equal to that guy. Angle AOB is 90 degrees. Angle AOB is 90 degrees. The way angle representation goes, the middle one is the point that they're talking about. So what you do, you start off from here and you go down. So AOB. So this guy is 90 degrees. If BO is 3 and AO is 4, BO is 3 and AO is 4. Now they want us to do solve AB, OQ and OW and QW. So there's four things that they want here from us. And this question will probably be worth two marks or four marks depending on how easy your teacher is because it's not a very hard question. So each one might be worth half a point, okay? Now, let's see what we can solve for first. They want solve for AB. So they want us to solve for this. Now, if we remember our Pythagorean theorem because it applies to right angle triangles, Pythagorean theorem says A squared plus B squared is equal to C squared. And that's what we have here. We have a right angle triangle. We have two sides of a triangle so we can always figure out the third side which happens to be the hypotenuse here. Now if they gave us this one and they wanted us to find this, your C would have been a number and your unknown would have been your A or B, okay? So don't get confused. C is always across from the 90 degrees. So right now we can solve for this. Let's take a look. This becomes 4 squared plus 3 squared is equal to C squared. Now 4 squared is 16 plus 3 squared is 9 is equal to C squared. 16 plus 9 is 25. So C squared, I'm just going to come down here, C squared is equal to 25 and to get C by itself you do the opposite of squared which is square root. So C is equal to 5. So we just solve for this guy, 5. Whatever you solve for something, if you want to circle it that way you know exactly what you solve for and these are the things that they gave you. So we've done this one. Do a little trick. Now I want to find out what OQ is. OQ. Now this one is going to be easy because they told us that this is equal to this. Where did they tell us this in the second part? O is the midpoint of BQ. So if this is equal to this then this has got to be 3. So we just solve for this one. Then they got OW. OW, well that's the same as this because of the first point they gave us. So if that was 4 then this has got to be 4. So we just solve for this one. Now there's two ways you can figure out what QW is. This length here. One way is doing the Pythagorean theorem again because if you remember your properties of angles, these two angles are vertically opposite. So if they're vertically opposite, if you have two lines crossing each other, okay, if you have two lines crossing each other, this angle is going to be equal to this angle. So if you ever have two lines going like this, this guy is always going to be equal to this and this guy is always going to be equal to this. This relationship is called vertically opposite. So what we have here, if that's 90 degrees then that's 90 degrees. Well if that's going to be 90 degrees, what we can do is use the Pythagorean theorem because A squared plus B squared is going to be equal to C squared, which is the same thing we did there. So this guy is going to be 5. The other thing, the other way we can do this is use the vertically opposite relationship and say yes, this is 90 degrees as well. Then what we have is congruent triangles because we have side angle side here and this is side angle side. So what you could say is point Q, QW is equal to AB because of side angle side and that's a legitimate answer. Therefore QW is equal to 5. So you wouldn't have to go through all these calculations if you knew your congruent triangles. Side angle side, side angle side. If you have three properties that are non-related in two triangles equal to each other, then the two triangles are congruent which basically means they're identical. If they're identical then this length must equal this length. So right there you prove that this was 5. So from this we just solved all four points. Now these types of questions they can state in multiple ways. They can give you triangles going all over the place and you have to start off in a certain area and you work your way through the whole problem, the drawing, to get all the answers. And the most important thing you can do with this stuff is visualize it. Transfer all the information they've given you here in the problem, in English form to a visual form. So you can actually see the patterns coming out at you. That way you can make appropriate marks to say what's equal to what and transfer the information over. And that's the point of one of the biggest properties associated with geometry, one of the biggest ways, one of the most important ways you solve geometry problems is lay it all out, transfer all the information over. Now keep in mind once you're done with the drawing it just becomes chaos. Now you better know you better be able to follow your your own work and be neat about it. Now I'm limited according to the space I have on this board to do this problem. You will be doing a piece of paper so make sure you organize all the information, okay. And whenever you get an answer put the information down. So over here we should have been going AB is equal to five, that's one mark. OQ is equal to three, that's two marks. OW is equal to four, that's three marks and QW is equal to five. And that's another reason why you circle your answers on the drawing. That way you don't have to go back and try to figure out what's what. Now we're going to do some more proofs regarding that stuff, okay.