Determining if Triangles Exist - Nerdstudy
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Let's learn to determine if a triangle exist
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There are often cases where we might be provided with ONLY the information, in terms of the dimensions and angles, of a triangle and must identify whether or not the given information can even construct a possible triangle in the first place. So if this sounded a bit confusing to you, stick around because in this video we’ll go through how to identify cases where a triangle can or cannot be made given limited information without any diagrams, and by the end of this video you’ll have a much better understanding and grasp of this concept.
So here is an acute triangle. If we were given ONLY the information to THIS angle, the length of the side next to it, AND the length of the side opposite to the angle without the actual diagram, then before we just go off using the sine law, we must first consider whether it is even possible to form a triangle with the provided information. For example, since this angle is defined, and these two sides would also be defined, if the length on the opposite side of the angle was defined to be shorter than the height of this triangle, then that would mean there is no triangle that exists. It would literally just look like this incomplete triangle we have here.
So let’s put some numbers into this to get a better idea of what we’re talking about. For example, if I said that there is an angle of 42 degrees, with the side length of 5 next to it, AND that the side length of the opposite side to the angle is 2, can we confirm that this would be a complete triangle? Well, this sounds awfully suspicious considering the fact that this doesn’t really look right to begin with. But there is something we can do to confirm whether the given information was for a complete triangle or not. What we can do is find the height of this triangle to see whether the value of THIS side is shorter than the actual height of the triangle. If it is, then we would know right away that this would not be a complete triangle since any dimension shorter than the height could not possibly connect to this side of the triangle.
Now how would we find the height of this triangle anyways? Well, since the height of a triangle is perpendicular to the base we know that this angle here is 90 degrees, making THIS a right angled triangle! And because we have the values of an angle and a side length we can use SOHCAHTOA to find the length of another side.
So, here we have the opposite side to the angle and the hypotenuse involved in this question, so we know we can use the Sine function. So let’s bring out the formula and start by plugging in our values. Multiplying both sides by 5 gives us THIS, so let's just rearrange this to get THIS. Finally, if we compute this, we get a final value of ‘h’ equals to roughly 3.35. So there we have it, since the length of THIS side is actually shorter than the length of the height, we know that the triangle would look more like THIS instead and that there is NO triangle created with the given information. Awesome!
Now, let’s turn our attention to a scenario where the angle in the triangle is not acute, but instead obtuse. If were given the angle and the side beside the angle is of length ‘b’, then the opposite side MUST be longer than ‘b’ or we would know for a fact that this triangle wouldn’t exist.
For example, if this angle here was 120 degrees, and this side next to the angle is 8 cm in length, then we just established that the side opposite to the angle must be greater than 8 cm at the very least. This is because, if side ‘a’ was exactly 8 cm as well, we would just end up having side ‘a’ and side ‘b’ overlapping each other, consequently creating this two lined figure. Therefore, side ‘a’ must be anything GREATER than 8, so even if it was 8.1, although it’s a very small difference, we’d see this super slim triangle created and thus, would at least be a completed triangle.
Great! So that’s pretty easy isn’t it? In our next video we’re going to look into a situation where we are given the dimensions of a triangle where it could possibly create 2 DIFFERENT triangles. We call this the ambiguous case.
But, that’s it for this lesson, make sure to keep practicing more questions and we will catch you guys in the next one.
Let's learn to determine if a triangle exist
--
There are often cases where we might be provided with ONLY the information, in terms of the dimensions and angles, of a triangle and must identify whether or not the given information can even construct a possible triangle in the first place. So if this sounded a bit confusing to you, stick around because in this video we’ll go through how to identify cases where a triangle can or cannot be made given limited information without any diagrams, and by the end of this video you’ll have a much better understanding and grasp of this concept.
So here is an acute triangle. If we were given ONLY the information to THIS angle, the length of the side next to it, AND the length of the side opposite to the angle without the actual diagram, then before we just go off using the sine law, we must first consider whether it is even possible to form a triangle with the provided information. For example, since this angle is defined, and these two sides would also be defined, if the length on the opposite side of the angle was defined to be shorter than the height of this triangle, then that would mean there is no triangle that exists. It would literally just look like this incomplete triangle we have here.
So let’s put some numbers into this to get a better idea of what we’re talking about. For example, if I said that there is an angle of 42 degrees, with the side length of 5 next to it, AND that the side length of the opposite side to the angle is 2, can we confirm that this would be a complete triangle? Well, this sounds awfully suspicious considering the fact that this doesn’t really look right to begin with. But there is something we can do to confirm whether the given information was for a complete triangle or not. What we can do is find the height of this triangle to see whether the value of THIS side is shorter than the actual height of the triangle. If it is, then we would know right away that this would not be a complete triangle since any dimension shorter than the height could not possibly connect to this side of the triangle.
Now how would we find the height of this triangle anyways? Well, since the height of a triangle is perpendicular to the base we know that this angle here is 90 degrees, making THIS a right angled triangle! And because we have the values of an angle and a side length we can use SOHCAHTOA to find the length of another side.
So, here we have the opposite side to the angle and the hypotenuse involved in this question, so we know we can use the Sine function. So let’s bring out the formula and start by plugging in our values. Multiplying both sides by 5 gives us THIS, so let's just rearrange this to get THIS. Finally, if we compute this, we get a final value of ‘h’ equals to roughly 3.35. So there we have it, since the length of THIS side is actually shorter than the length of the height, we know that the triangle would look more like THIS instead and that there is NO triangle created with the given information. Awesome!
Now, let’s turn our attention to a scenario where the angle in the triangle is not acute, but instead obtuse. If were given the angle and the side beside the angle is of length ‘b’, then the opposite side MUST be longer than ‘b’ or we would know for a fact that this triangle wouldn’t exist.
For example, if this angle here was 120 degrees, and this side next to the angle is 8 cm in length, then we just established that the side opposite to the angle must be greater than 8 cm at the very least. This is because, if side ‘a’ was exactly 8 cm as well, we would just end up having side ‘a’ and side ‘b’ overlapping each other, consequently creating this two lined figure. Therefore, side ‘a’ must be anything GREATER than 8, so even if it was 8.1, although it’s a very small difference, we’d see this super slim triangle created and thus, would at least be a completed triangle.
Great! So that’s pretty easy isn’t it? In our next video we’re going to look into a situation where we are given the dimensions of a triangle where it could possibly create 2 DIFFERENT triangles. We call this the ambiguous case.
But, that’s it for this lesson, make sure to keep practicing more questions and we will catch you guys in the next one.
8 سال پیش
در تاریخ 1395/11/23 منتشر شده
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