Determining triangle uniqueness
About this video
A triangle has a 40 degree angle, a 6 cm side opposite that angle, and another side of 8 cm. By sketching, the narrator first shows at least one such triangle exists, then opens the shape up to draw a second, different one (one with an acute angle, one with an obtuse angle), so more than one triangle fits the description.
You'll be able to
- sketch triangles from given angle and side conditions.
- decide whether conditions give no triangle, exactly one, or more than one.
- explain how the same measurements can make two different triangles.
Key words
- unique triangle
- The only triangle that fits a description; any other drawing would be an identical copy.
- opposite
- Across from; the side opposite an angle does not touch that angle.
- acute angle
- An angle smaller than 90 degrees.
- obtuse angle
- An angle bigger than 90 degrees but smaller than 180 degrees.
Transcript
- [Instructor] A triangle has an angle measuring 40 degrees, a 6-centimeter side opposite the 40 degree angle, and another side that is eight centimeters long. How many unique triangles can be drawn to fit this description? It's either more than one, exactly one, or no triangle.
So pause this video. Have a go at it before we do it together. Okay, well, I'm just gonna see how many triangles I can draw. (chuckles) So let's see. If we have a triangle, well, I'm gonna to do a 6-centimeter side here. Let's say that this right over here is six centimeters.
We'll just assume everything's in centimeters, actually. And let's see, it has another side that is eight centimeters. So I'm gonna try to draw it to scale, but it's not gonna be perfectly at scale. So if that is six, eight centimeters might be something like this.
So this is the 8-centimeter side. And then there is this angle right over here that they're telling us is an angle measuring 40 degrees. And they said that that is, that the six centimeters is opposite the 40 degree angle. So I already know that I can construct this triangle.
In fact, it seems very reasonable. It's not that one side is longer than the sum of the other sides or somehow that all of my angles would have to add up to more than 180 degrees, which is impossible. So this could be a very reasonable angle, a very reasonable triangle.
I don't know what this last side is, but I can definitely rule out no triangle. So let me see if I can think of another way to draw a different triangle. So I'm gonna open it up even more. So something like this. And a 40 degree angle, and I'm just sketching it right here.
So this right over here, this could still be a 40 degree angle. This could still be an eight side. I've already drawn two different scenarios, one where I have an acute angle here, one where I have an obtuse angle here, but in either case, I could have the 40 degree angle or the 6-centimeter side be opposite of the 40 degree angle.
So I like more than one triangle.
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