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USING STRUCTURE ind the circumference of each circle.a. a circle circumscribed about a right triangle whose legs are 12 inches and 16 inches longb. a circle circumscribed about a square with a side length of 6 centimetersc. a circle inscribed in an equilateral triangle with a side length of 9 inches

$\mathrm{C}=20 \pi$

Geometry

Chapter 11

Circumference, Area, and Volume

Section 1

Circumference and Arc Length

Area and Perimeter

Surface Area

Volume

Circles

Johns Hopkins University

Missouri State University

Cairn University

University of Michigan - Ann Arbor

Lectures

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Hey, this is problem. 34 part, eh? We have a circle and we have a triangle inside of that circle. The points of the circle are a we got to see and we got B A C A is also the diameter of that circle. We know that a B is 16 inches. BC is 12 inches. So now we need to find out what the like the diameter is to do that use the Pythagorean theorem We're gonna go a C squared equals a B squared plus b sea squirt. So we're gonna have a C squared equals 16 squared plus 12 squared a C squared that is going to be 400. That means that a C has a length of 20 inches and that's the diameter. We know that the circumference is the same as saying two pi r. We also know that the diameter is the same as two times the radius. So we know that the circumference can be the diameter. Times are so Excuse me the diameter times pi not R. So we can substitute in 20 for the D for the diameter and our circumference is 20 pie which is approximately 62 0.83 inches. Que part me. We have a square inside of a circle and that square the side is six centimeters. So we cut. If we draw a line from corner to corner, that's gonna be I don't our diameter. So this is also the diameter of that circle. We can use it with a degree in there also. So again, if this is a B c N d, we know that a C squared is gonna equal a D squared plus d c squared. So a C squared is going to equal the six squared plus six squared, which is equal than 36 plus 36 which is he called a 72. So a C is gonna be the square root of 72 which is equal to six times the square root of two. And this is centimeters so we can take there's a conference equals two pi r We know that the diameter equals two are so we also know that the circumference can be I don't but diameter times pi. So if we substitute that in, we're gonna take six escorted to a multiply that by pi and we calculate that out. That's gonna be approximately 26.664 centimeters apart. See, we have unequal lateral triangle. You have a circle in the middle. We have a radius that when drawn down, will cut that side in half. We know that each side on that equal angle triangle is nine inches, so this is 44.5 inches. So the one side of this triangle we're gonna draw, we're gonna draw a triangle inside of the circle. We're gonna go from the center, and we're gonna cut this angle into when we cut that into because unequal lateral triangle has 60 degrees for each of its angles, that angle is going to be 30 degrees. We also know that this is a right angle. So this here is 90 Digger Grace. So what we have left is this one here is 60 degrees. So knowing that we're gonna use what they call the 30 60 90 there, um, that tells me in the 30 60 90 here I am. The hype oddness is equal to two times the short leg. I've been breathing it, abbreviating that s o. We now know that the longer leg, which I'm showing us l is equal to the shorter leg times the square root of three. So what we need to do is we need to figure out what the length of our heIp heip oddness is, which is blue line that I've drawn there. We can find that then and actually, we don't need to do that. I'm sorry. Um, best thing we can do is we know the long legs 4.5 inches equals the short leg times the square root of three. If we divide by the square root of three, both sides that's gonna give us, um 4.5 over the square root of three is equal to a short length the short leg. So if we take both of these times the square to three, we're gonna end up with 4.5 the square to three all over three, which is sequel. The 1.5 square root of three. And that's our short leg. Okay, so now the circumference it's gonna be to pie r r r is a short leg, so we're gonna say two pie times, 1.5 square root of three. We multiply all of that out, and we're going to get 16 0.32 inches

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