3) Pedro is constructing a triangular banner with a blue border on one side; a yellow border on one side; and a red border 0n the remaining side The angle between t...

The frame of a kite consists of six pieces of lightweight plastic. As shown in FIGURE $4.8 .33,$ the outer frame of the kite consists of four precut pieces; two pieces of length $2 \mathrm{ft}$ and two pieces of length 3 ft. The remaining crossbar pieces, labeled $x$ in the figure, are to be cut to lengths so that the kite is as large as possible. Find these lengths.

So we know that little a. Is supposed to be 3" long, Side B. Is longer, it's four inches long and angle alpha opposite side A. Is supposed to be 30 degrees. And so if we kind of draw this triangle out and so here is angle alpha. And there's my 30 degree angle. And let's say this is side B. And it's four inches. And then this side opposite, let's say it's 3". Now we can see that we have short side alongside angle, short side alongside angle and again this is side little A. So do we have no triangle, one triangle or two triangles? And if we have, we first of all want to determine what this altitude would be. Let's assume we can draw the triangle And maybe this altitude H. is actually equal to a. And maybe this is supposed to be a three right here. So let's determine if that what that help altitude would have to be. And that would be the sign of 30°. What equal the opposite side over the high partners in that triangle. And sign of 30 degrees is equivalent to a one half. That means the altitude would have to be two inches. So this altitude would be two inches. And then this side a we can draw and in fact we can draw two triangles. One triangle would be as if I tell my students take like a compass and think about this being you know, four inches long and now this guy is three inches long. And if I put the pointing under the compass here and then stretch that out to three inches. There's one triangle where I'm going to end up having uh this it looks like. So and then I could take the pointy part, the pencil part and come over here. And I could also have A triangle where this is 3" right here and I'd have this obtuse triangle. So I'd have an acute triangle and an obtuse triangle. So I have that my A. Measurement is bigger than the height and it is also the height is it's larger than not that be, so it's smaller than that, that letter B. It's in that setting. So anytime this happens where I have my lips, I just wrote the wrong way, I have my A. Being larger than my height, but I'm having it to be smaller than that side B. And that will give me that short side alongside angle. And I will have two triangles.

So we're given a triangle with a being six Be being four and Alpha being 30 three's. And so if this is angle alpha And it's 30° and we have the side opposite as six inches Would say that that is 6". And then that is little A. And let's say that this is side b. And that is 4". Then we can see we actually have long side, short side angle. And that is a unique triangle. And notice that this angle would have to be smaller then 30 degrees because it's opposite a shorter side. And so when you would add these two together, the some of these two is going to be less than Less, less than 60° making this angle and um to strangle. So we will have to have an obtuse triangle, an obtuse triangle. And it will be a unique obtuse triangle. So one triangle.

So these legs are 30 inches long, but I need them to be 26 3/4 inches long. So what operation am I going to do? What is this different change here? Well, I'm probably going to subtract thes to to figure out how much you need to cut off. So with subtracting mixed numbers, 30 minus 26 will give me four. And then, technically, over here, I have 30 and 4/4. Right? So just a regular amount minus 3/4 will be one fourth of an inch. Now, the difference here is technically toe have 30 inches. 3/4. This is actually going to be, um, same thing as, um 29 4/4 is the same thing as 30 inches and 3/4. So, actually, what this will be is 29 minutes, 26 is three 4/4 minus 3/4 will be 1/4 s. So that is how much I will have to cut off

Were given two sides and angle internal to them. We're trying to figure out how Maney Solutions exists that fit that description for our triangle. And we're looking at a table to in the book to determine that we are going to need to figure out the height of this triangle first. So that formula is be sign Alfa. In this case, that's four inches times the sign of 30 degrees or two inches. One of the descriptions we have for a number of solutions is when H is less than A and A is less than B. In this case, age is too. Inches a is three inches, B is four inches, so that is true for this triangle. And if that happens to be the case, there's actually two triangle solutions that work.