Understanding the Mid-Segment Theorem

In this lesson we're going to be talking about the mid-segment theorem that pertains to triangles. So in order to understand the mid-segment theorem, what I want to do is make a point that is going to be halfway between A and B. So let's say that is going to be right here. This is the midpoint of this line segment AB. So this point is going to split AB, this entire line AB, into two equal sized parts. Then we can do the same thing with this line segment BC. Let's find the midpoint of this line segment BC and that's going to be over here. And that point is going to split BC into two equal sized parts. Now the interesting thing about the mid-segment theorem is that it says that if we were to draw a line between these two midpoints, so we would go from this midpoint to this one. So this line is joining the midpoints of A, B, and B, C. And what the mid-segment theorem tells us is that when we have a line joining the midpoints of two sides of our triangle, that line is going to be parallel to the other side, and it is going to be half the length of the other side. So the other side that we're referring to is the side that is not included. So we can see that in this case, the mid-segment, which is this line over here, is connecting... this line and this line. So AC is the line that is not involved and that means that this mid-segment will be parallel to that side, that is the other side. So this mid-segment here is going to be parallel to this line AC. It is also going to be half the length of AC. So if we were to split AC into two equal parts, that is if we were to include a mid- like we've done for AB and for BC, if we were to do that for AC that midpoint would be over here and since this mid segment is half the length of this line AC we know that whatever this length is, let's say this has this length, we know that this side is going to have that length and this side is going to have that length because here this midpoint of AC is going to split AC into two halves. So here we have one half and here we have one half and since this mid segment here is half of the length of AC It's going to be the same as from C to this midpoint and from A to this midpoint And what I'm just going to do is label some more points in our triangle So let's label this D and we can label this E And I want to label this point S. So what we have just seen is that DE is going to be parallel to AC Line segment DE will be parallel to line segment AC, and the line segment DE is going to be congruent to AF, and DE is going to be congruent to CF. So that is basically the mid-segment theorem. It's that if you join the midpoints of two sides of a triangle with a line, that line is going to be parallel to the other side, and will be half the length of that other side. So let's do the same for these two points now. Let's join points D and points F. So now what we're doing is we're joining the midpoints of this line segment AB and this line segment AC. So if we were to join the midpoints of these two sides this time, we're going to get a line that looks like that. And now DF is going to be parallel to the other side and what's the other side? Well that's going to be the side that we did not join together so that's going to be this side BC. So DF this mid segment that we joined from AB and AC is going to be parallel to BC and it will be half the length of BC. So DF here is going to be parallel to BC and its length is going to be half the length of BC and we can see that this point E breaks up BC into its two halves. So that means that the length of DF is going to be the length of BE and the length of CE. So let's say that is going to be one red notch, that's going to be one red notch. DF is also going to be one red notch. So basically we show that DF, which is this mid-segment here, is going to be parallel to BC And DF is going to be congruent to BE and DF is going to be congruent to CE. We can also do the same thing joining these two midpoints together. So now we're going to be joining the midpoint of the side AC with the midpoint of the side BC. If we were to do that, that's going to look like this. This midsegment, FE, is going to be parallel to the other side. And what is the other side again? The other side will be the side that you didn't join using your two midpoints. So the side that's going to be parallel to this mid segment Fe is this side, AB. So Fe, which is our mid segment, is going to be parallel to AB. And Fe is going to be half the length of AB. So this midpoint D is going to separate AB into these two halves, and each of these halves will be the same length as Fe. So if... AD has this value, then DB is also going to have that value, and FE will have that same value. It is also going to be parallel to AB. So we know that the length of FE is going to be congruent to AD, and the length of FE is also going to be congruent to DB. So that is going to summarize the mid-segment theorem. Basically if you are going to make points that are halfway between your line segments of each of the sides of your triangle, if you join the two midpoints of two of your sides together, the line that you create by joining the two midpoints of two of your sides of your triangle is going to be parallel to the third side of the triangle and will be half the length of that third side of the triangle. And knowing that is going to allow you to solve some problems when it comes to triangles like this. So let's say you were given a triangle that looked like this. So you were given this triangle and what we can already tell is that each of these points D, E, and F are going to be midpoints because as we can see each of these points splits the entire line into two equal parts. So BF is equal to FC, CE is equal to AE, AD is equal to BD. So we know that these are midpoints. And let's say this said if DE is equal to 30, what is BC? So we know here that DE has a value of 30. And what we know via the mid-segment theorem is that since this mid-segment is joining sides AB and sides AC, The mid segment will be parallel to the other side which is side BC and it will be half the length of that side. So this side DE is going to be half the length of BC. In other words, BF is going to be equal to 30 and FC is also going to be equal to 30. So that means that BC is equal to 2 times DE which means that BC is equal to 2. times 30 which is 60. So that's an example of how you can use the mid-segment theorem to answer certain questions. You might get a case in which you're given one of these side lengths and then you're asked to find the length of one of the mid-segments and you can use these same relationships to calculate that. You might have a case in which you are given the length but you're not given it a full number, it's given in terms of an equation. So let's say I said that the length of DE was 2x plus 5, and told you that BC was 5x plus 2, then you can use simultaneous equations knowing these relationships between the sides and you can easily calculate questions like that.

So this point is going to split AB, this entire line AB, into two equal sized parts. Then we can do the same thing with this line segment BC. Let's find the midpoint of this line segment BC and that's going to be over here. And that point is going to split BC into two equal sized parts.

Now the interesting thing about the mid-segment theorem is that it says that if we were to draw a line between these two midpoints, so we would go from this midpoint to this one. So this line is joining the midpoints of A, B, and B, C. And what the mid-segment theorem tells us is that when we have a line joining the midpoints of two sides of our triangle, that line is going to be parallel to the other side, and it is going to be half the length of the other side. So the other side that we're referring to is the side that is not included.

So we can see that in this case, the mid-segment, which is this line over here, is connecting... this line and this line. So AC is the line that is not involved and that means that this mid-segment will be parallel to that side, that is the other side.

So this mid-segment here is going to be parallel to this line AC. It is also going to be half the length of AC. So if we were to split AC into two equal parts, that is if we were to include a mid- like we've done for AB and for BC, if we were to do that for AC that midpoint would be over here and since this mid segment is half the length of this line AC we know that whatever this length is, let's say this has this length, we know that this side is going to have that length and this side is going to have that length because here this midpoint of AC is going to split AC into two halves. So here we have one half and here we have one half and since this mid segment here is half of the length of AC It's going to be the same as from C to this midpoint and from A to this midpoint And what I'm just going to do is label some more points in our triangle So let's label this D and we can label this E And I want to label this point S.

So what we have just seen is that DE is going to be parallel to AC Line segment DE will be parallel to line segment AC, and the line segment DE is going to be congruent to AF, and DE is going to be congruent to CF. So that is basically the mid-segment theorem. It's that if you join the midpoints of two sides of a triangle with a line, that line is going to be parallel to the other side, and will be half the length of that other side. So let's do the same for these two points now.

Let's join points D and points F. So now what we're doing is we're joining the midpoints of this line segment AB and this line segment AC. So if we were to join the midpoints of these two sides this time, we're going to get a line that looks like that.

And now DF is going to be parallel to the other side and what's the other side? Well that's going to be the side that we did not join together so that's going to be this side BC. So DF this mid segment that we joined from AB and AC is going to be parallel to BC and it will be half the length of BC.

So DF here is going to be parallel to BC and its length is going to be half the length of BC and we can see that this point E breaks up BC into its two halves. So that means that the length of DF is going to be the length of BE and the length of CE. So let's say that is going to be one red notch, that's going to be one red notch.

DF is also going to be one red notch. So basically we show that DF, which is this mid-segment here, is going to be parallel to BC And DF is going to be congruent to BE and DF is going to be congruent to CE. We can also do the same thing joining these two midpoints together.

So now we're going to be joining the midpoint of the side AC with the midpoint of the side BC. If we were to do that, that's going to look like this. This midsegment, FE, is going to be parallel to the other side. And what is the other side again? The other side will be the side that you didn't join using your two midpoints.

So the side that's going to be parallel to this mid segment Fe is this side, AB. So Fe, which is our mid segment, is going to be parallel to AB. And Fe is going to be half the length of AB. So this midpoint D is going to separate AB into these two halves, and each of these halves will be the same length as Fe. So if...

AD has this value, then DB is also going to have that value, and FE will have that same value. It is also going to be parallel to AB. So we know that the length of FE is going to be congruent to AD, and the length of FE is also going to be congruent to DB.

So that is going to summarize the mid-segment theorem. Basically if you are going to make points that are halfway between your line segments of each of the sides of your triangle, if you join the two midpoints of two of your sides together, the line that you create by joining the two midpoints of two of your sides of your triangle is going to be parallel to the third side of the triangle and will be half the length of that third side of the triangle. And knowing that is going to allow you to solve some problems when it comes to triangles like this. So let's say you were given a triangle that looked like this. So you were given this triangle and what we can already tell is that each of these points D, E, and F are going to be midpoints because as we can see each of these points splits the entire line into two equal parts.

So BF is equal to FC, CE is equal to AE, AD is equal to BD. So we know that these are midpoints. And let's say this said if DE is equal to 30, what is BC? So we know here that DE has a value of 30. And what we know via the mid-segment theorem is that since this mid-segment is joining sides AB and sides AC, The mid segment will be parallel to the other side which is side BC and it will be half the length of that side.

So this side DE is going to be half the length of BC. In other words, BF is going to be equal to 30 and FC is also going to be equal to 30. So that means that BC is equal to 2 times DE which means that BC is equal to 2. times 30 which is 60. So that's an example of how you can use the mid-segment theorem to answer certain questions. You might get a case in which you're given one of these side lengths and then you're asked to find the length of one of the mid-segments and you can use these same relationships to calculate that.

You might have a case in which you are given the length but you're not given it a full number, it's given in terms of an equation. So let's say I said that the length of DE was 2x plus 5, and told you that BC was 5x plus 2, then you can use simultaneous equations knowing these relationships between the sides and you can easily calculate questions like that.

Transcript for:Understanding the Mid-Segment Theorem

Transcript for:
Understanding the Mid-Segment Theorem