Learn about Segment Addition In this lesson, let’s learn how to add segments and learn about this segment addition postulate. We’ll do three problems but the fundamental thing that we learn is if we have a line, a line segment, which is A, B, and C, and we say that B is in between A and C, then the length of AB plus the length of BC equals the length of AC, which should make commonsense, right? The length of this segment plus the length of this segment is the length of the entire segment, which visually makes sense to me. This is called a segment addition postulate and this only applies if B lies on the same line as AC. So let’s use the segment addition postulate to solve three problems. First, it tells us C is between A and D. We need to find the length of AD. What do we know? Well, we know AD = 7x. We also know AC = 2x + 7 and CD = 33. So based on the segment addition postulate, AD should be equal to AC + CD because this C lies between A and D. So if this is true, this implies AD is 7x = AC, which is 2x + 7 + CD, which is 33. The sum of these two should equal the total line. So this is my equation. How do I solve it? We have 7x here, 2x here, subtracting 2x from both sides. What do I get? 5x equals—this gets 0, 7 + 33 = 40, which implies x = 8. So if x is 8, what is AD? AD is 7x, so AD = 7(8) which is 56. Remember, all we did is we understood that the length of AD is 7x and the length of the two parts that make it up are these. These two must be equal and it gives us an equation. Once we have an equation, we solve it to get the value for x. Once we know x, we know what AD is, 7 times x, which is 56. Let’s apply the same principle to the second problem. It says B is the midpoint of AC. Midpoint means this distance and this distance are exactly the same. AB = 5x and BC = 3x + 4. Find AB, BC and AC. What am I told? If this is the midpoint, that means this equals this, so 5x = 3x + 4, again, subtracting 3x from both sides, what do I get? 2x = 4 or x = 2. If x =2, what is AB? 5 x 2 = 10. What is BC? Three times two, 3x, 3 x 2 = 6 plus 4 is 10. It makes sense because AB and BC are supposed to be equal. And what is AC? Well, that’s going to be the sum of both of these, right? AC is the sum of both of these, so 10 + 10 = 20. That’s all we have to do. Remember that if a point is exactly the midpoint between the two, both the lengths are equal. Let’s try one more, slightly more difficult. It says the map shows the route for a race. This is the route. Start point is 365 meters from a drink station R, so from here to here, it’s 365 meters, and 2 kilometers from drink station S. So from here to here, it’s 2 kilometers or 2000 meters; one kilometer is 1000 meters. First aid station is located at the midpoint of the two drink stations. So this and this are the same because the first aid station is exactly between the two drink stations. How far is the start point? So let’s do this. What do we know? We know line segment AR, A to R is 365. We know A to S is 2000, all the way to the drink station S is 2000. We know that RF = FS because F is the midpoint. What else can I deduct from what I’ve been given? Well, if this is 2000 and this is 365, then this part is 2000 – 365. So RS = 2000 – 365, which is 1635 meters. So that’s what we’ve been given, so here’s what we do. We know RF + FS = 1635 but we also know these two are equal, which means RF = FS equals 1635/2, which is 817.5 meters. How did we do that? Well, if RF and FS are the same and they total 1635, then each one of these is half of 1635. So we know this is 1635/2 or 817.5. Now that we’ve measured all of these, how far is the start point from the first aid station? We need this distance. Well, what is it? It’s this plus this, so what is AF? AF, which is what we’re looking for, is 365 + 817.5 in meters, which is 1182.5 meters. The start point distance from the first aid station is 1182.5 meters. And again to repeat, what we did is we just learnt about this segment addition postulate, w