Question Video: Finding the Length of the Midsegment in a Given Triangle Mathematics

If the perimeter of โ–ณ๐ด๐ต๐ถ = 9.7 cm, ๐ธ is the midpoint of the line segment ๐ด๐ถ, and the line segment ๐ท๐ธ โˆฅ the line segment ๐ต๐ถ, find the length of the line segment ๐ท๐ธ.

03:23

Video Transcript

If the perimeter of triangle ๐ด๐ต๐ถ is equal to 9.7 centimeters, ๐ธ is the midpoint of the line segment ๐ด๐ถ, and the line segment ๐ท๐ธ is parallel to the line segment ๐ต๐ถ, find the length of the line segment ๐ท๐ธ.

So the first thing we can do is use a bit of information that weโ€™ve been given. And that is that the perimeter of the triangle ๐ด๐ต๐ถ is equal to 9.7 centimeters because what we can do is use this to help us calculate the length of the line segment ๐ต๐ถ. Because what we can say is that the line segment ๐ต๐ถ is going to be equal to the value we get when we subtract the lengths of the two sides away from the perimeter. So what weโ€™re gonna have is 9.7 minus 3.1 minus 2.9, which is equal to 3.7. So therefore, we can say that the line segment ๐ต๐ถ is 3.7 centimeters.

Okay, great. But what do we do next? Well, if we take a look at our diagram, we can see that weโ€™ve got these lines marked on at ๐ด๐ธ and ๐ธ๐ถ, so our two line segments there. So therefore, what we can say is that line segment ๐ด๐ธ is equal to line segment ๐ธ๐ถ. So therefore, we can say that if we want to define line segment ๐ด๐ธ, well, this is going to be equal to 2.9 divided by two, which is gonna be equal to 1.45 centimeters. So therefore, we can also say that the line segment ๐ธ๐ถ is going to be equal to 1.45 centimeters.

So at this point, weโ€™re thinking, โ€œOkay, how does this help us to find the length of the line segment ๐ท๐ธ?โ€ Well, what we can start to look at now is something called the side splitter theorem. What the side splitter theorem tells us is that if a line is parallel to a side of a triangle and the line intersects the other two sides, then the line divides those sides proportionally. But what does this mean in practice? Well, letโ€™s look at the sides of our triangle.

Well, for our triangle, this tells us that ๐ด๐ธ over ๐ด๐ถ is equal to ๐ด๐ท over ๐ด๐ต, which is equal to ๐ท๐ธ over ๐ต๐ถ. So we can see here that Iโ€™ve put our fractions this way round, so Iโ€™ve have put the shorter sides on the top and the longer sides on the bottom. It can actually work the other way around just as long as we keep this consistent. Okay, great. So how can we use this to help us solve our problem? Well, as weโ€™re trying to find the length of the line segment ๐ท๐ธ, we can say that ๐ด๐ธ over ๐ด๐ถ, and thatโ€™s because we now know ๐ด๐ธ and we know ๐ด๐ถ, is equal to ๐ท๐ธ over ๐ต๐ถ. So we can say that 1.45 over 2.9 is equal to ๐ท๐ธ over 3.7.

So therefore, we can carry on our calculation over here. And weโ€™re gonna have 0.5 is equal to ๐ท๐ธ over 3.7. So then, if we multiply both sides of the equation by 3.7, weโ€™re gonna have 0.5 multiplied by 3.7 equals ๐ท๐ธ. So therefore, weโ€™re gonna get ๐ท๐ธ is equal to 1.85.

So therefore, we can say that in answer to the question, the length of the line segment ๐ท๐ธ is equal to 1.85 centimeters. And we found that using the side splitter theorem.

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