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TRUSSES $Q$ is the midpoint of the line segment $\overline{PR}$ in the truss rafter shown in the figure. What are the lengths of the line segments $\overline{PQ}$, $\overline{QS}$, and $\overline{RS}$?

$P \overline{Q}=9.43, \overline{Q S}=5, \overline{R} S=12.8$

01:41

Wen Z.

Precalculus

Geometry

Chapter 6

Additional Topics in Trigonometry

Section 2

Law of Cosines

Trigonometry

Introduction to Vectors

Area and Perimeter

Piedmont College

Cairn University

Boston College

Utica College

Lectures

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drop picture of this situation, first of all, so try to draw what kind of looks like a I saw a sleaze triangle there, and this is labeled are up here and then this comes straight down on a perpendicular, and then this is supposed to split this in half. This is supposed to be 88 doesn't say eight inches. You know, assuming that that's probably what it is, if it's a trust. And so all those air eight inches, this is labeled P, and then we have a point Q. Here. That's supposed to be the midpoint of this segment. And if we drop that straight down, there's another one. That is, uh, over here that they've drawn and let's see, what else do we know? We know that from here, let me just kind of doubt that over from here to here is 10 or probably 10 inches again, and we wanna find I don't think I have a something labeled I don't have s labeled, so we wanna find P Q. How long P Q is? We wanna find how long Q. S s and we want to find how long RS is, and we have a number of different ways, and again this would be on a perpendicular. But if I look at this triangle right here, this triangle, this red triangle, it's a right triangle and notice that this blue triangle is similar. And so we know how long this side is. This is 10 units long, and we do want to find how long this qs is. So we can say that in the Red Triangle that Qs is to eight. Leg is toe leg as this 10 is to 16 and then sobbing. That proportion are looking at this. We know that we have to If we go backwards 16 divided by two is eight sub 10, divided by two. This value must be five. So we could just use proportionate reasoning there. Um, now we know that this one is five. So we've just found this guy. We know that this one, we've just calculated to be five. Now, look, we wanna find P Q that looks like Pythagorean theorem to May. So we know that leg squared plus legs squared, square rooted will equal high pot, new square. So that looks like the square root of 89. And if I get a decimal approximation. That's about 9.43 So we know this one is 9.43 Now, if we wanted PR, we don't. But if we wanted PR, all we would have to do is double that length to find PR. Now, let's see what else we wanna find. We wanna find r s. Let's see, Rs is right here from here to here. Now look again. I have a right triangle and I wanna find this length and I can use Pythagorean theorem again. I know that eight squared or 64 plus 10 squared, which is 100 square rooted. Well, give me the high pot news. So I have the square root of 164 and if I get a decimal approximation for that, it's about 12.8. So we actually could do this with both agree and the're, um and some proportional reasoning. We could also do some things with Trig and find some other angles and use some of our other the're ums. But why go difficult when we could go simplistic, so similar triangles that works wonderfully with doing a little proportion and Pythagorean theorem

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