using-structure-ind-the-circumference-of-each-circle-a-a-circle-circumscribed-about-a-right-triangle

Question

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

Georgiann Andersen · Accepted Answer

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&#x27;re gonna go a C squared equals a B squared plus b sea squirt. So we&#x27;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&#x27;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&#x27;s gonna be I don&#x27;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&#x27;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&#x27;t but diameter times pi. So if we substitute that in, we&#x27;re gonna take six escorted to a multiply that by pi and we calculate that out. That&#x27;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&#x27;re gonna draw, we&#x27;re gonna draw a triangle inside of the circle. We&#x27;re gonna go from the center, and we&#x27;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&#x27;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&#x27;ve been breathing it, abbreviating that s o. We now know that the longer leg, which I&#x27;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&#x27;ve drawn there. We can find that then and actually, we don&#x27;t need to do that. I&#x27;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&#x27;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&#x27;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&#x27;s our short leg. Okay, so now the circumference it&#x27;s gonna be to pie r r r is a short leg, so we&#x27;re gonna say two pie times, 1.5 square root of three. We multiply all of that out, and we&#x27;re going to get 16 0.32 inches

Amrita Bhasin · Answer

as you can see over here, we see that from the center point. What&#x27;s named this? Oh, we can see a right triangle that&#x27;s inscribed. You can see the right angle over here indicating that it&#x27;s a right triangle. Then for part B. You know, the midpoint of the high pot noose is Thesent er of the circle that have partners is the longest side like this. The mid point. The point in the middle is the centre. In other words, point of for parts seat. The center of the circle lies in the midpoint of the high pot. Nous Okay, moving on to part D. We know the length of the high pot Innis two times the radius. Therefore, the pentagram here, um c squared equals a squared plus B squared. You know, this is 100. So we know that two hours the skirt of 100 which is 10 therefore Rs five because of tourist tendon Rs five

Julian Wong · Answer

this problem. We have a 64 sided polygon inside a circle. Um, so I&#x27;ve drawn It&#x27;s pretty hard to draw 64 sided shape, but this inside dotted line is supposed to represent that, and we&#x27;ve got a circle on the outside. It doesn&#x27;t matter what the entire shape looks like. We&#x27;re just gonna focus on one of these sections. So, uh, if this is one of the sides in the 64 sided shape, we are trying to find this wedge. We&#x27;re trying to find me radius of the circle. So I&#x27;m gonna just draw this a little bit bigger so that we can, uh, fill some numbers and you&#x27;ll have the side of the shape you&#x27;ll have the amount of Ah, the radius on the side. That&#x27;s our That&#x27;s our We know. The side of the shape is three. And of course, it&#x27;s got on our Klink because it&#x27;s bounded by a circle. Not that the circle matters in this first step. In part A. They&#x27;re asking us to find the radius of the circle. And, uh, well, what do we know about this shape? We only need the triangle. We actually have this angle faded This angle. Theta is your full 360 degrees divided by 64 slices. So first of all, let&#x27;s find the angle. Data is equal to 360 degrees divided by 64 slices. So we&#x27;re saying each slice as, ah, an angle of five points, 6 to 5 degrees. And our job is to find ah radius. So so in any triangle. And I&#x27;m gonna get rid of the our complaint here for a second in any triangle. Uh, the three angles should add up to one eighties. I&#x27;m gonna use different this angle. This angle and the angle data should add up to 1 80 recognizing that the two R&#x27;s are going to be the same. This is a nice sauce, Elise trying So what we can say and we&#x27;ll call this X X plus X plus data is equal to 180 we can call those exes because we know they&#x27;re the same due to ah, property from the ice off of these trying, the two x plus 5.6 to 5 equals 1 80 and we can put this into a calculator. 1 80 minus 5.625 divided by two gives us an X value of 87.1875 The last step is to find our itself, and we do this through the sign law. So, uh, the sign Law says, sign a over side A equals sign, be oversight beat where their angles and sides have to be a correspondent for the office, it of each other. So we can set this up as a sign of 5.6 to 5. The side opposite of that angle is three equals sign of, uh, 87.1875 and the side opposite to that is our so cross multiplying. Here we have our equals three times sine 87.1875 degrees over sign, 5.6 to 5 degrees, and that gives us an are of 30.6 centimeters. I&#x27;m rounding too. 36 things in Part B were asked to find the difference between the circumference of the circle and the perimeter of the polygon. So the difference D is equal to circumference of the circle is calculated by two pi r. We just found our so that&#x27;s great. And the perimeter of the 64 sided shape would be 64 times three. Because we have ah, regular. Probably gone. That&#x27;s gotta like three. And they&#x27;re 64 sides altogether. So when we put in 30.6 into our and take away 64 times three, this will give us a difference of 0.771 centimeters.

Wen Zheng · Answer

the problem is even a triangle with AIDS. You got to turn five is equal to fifty five and see is you go to seventy to find areas of the triangle. Second, scrap the circle and inscribed the circle. You&#x27;re really not area of the triangle is t Readers of the circumscribed circle is upto ar. The videos have been scrapped. The circle is smart. The first of the compute as is it going to be a plus B plus C over too. Which is it going to? Seventy six says the area of the triangle. He is he going to squire it of asked hams as malice, eh? I have asked Manus being halves as mine See. Is he going to five hundred and seventy on six. Zero B by the results of exercise sixty seven half you are. Is he going to a over sign off, eh? So now the area of a triangle is equal to one half ham. Speed him. See hams. Sign off, eh? Society, eh? Is there going to twenty over the sea? From this? We have to R is he go to a over two tea ham specie. So are you. It&#x27;s a call to a high speed have c over or tea which is the control you twenty five times fifty five seventy two over or about hundred seventy point six zero. Which is it true? Forty three point three eight is the area after circumscribed circle is equal to hi. Ah Squire, which is it going to? Off mine. Long long point nine three. I see. But the results have exercised sixty seven we have smart. Is he go to area of a triangle over us which is? They&#x27;re going to five hundred seventy point sixty year old over seventy six which is equal to seven point About one this area Is he going to heIp are square Which is it going to? One seven seven point one nine.

Lily An · Answer

So for this one, well, it&#x27;s going to first start out by drawing thes lines that we could make the square inside of here. So it tells us that the area which is a and the perimeter of B which is be off this square right here are the same. So a is equal to be so is going to write out the area. So we know that the area is going to be length squared because it&#x27;s a square. So all the links to the same and for the perimeter, it&#x27;s going to be four out because we&#x27;ve got four of these, then we&#x27;re just adding them together. So helpless, helpless, helpless. L it&#x27;s just foil. So in order to see what that&#x27;s going to be, we can go ahead and divide both sides, both sides by El and so we get l is equal to four. So now that we know that each of these lengths are for Let&#x27;s go in And look at this try well right here in isolation. So this struggle right here, you know, that&#x27;s a right angle, and this is for and both this right here and this right here are both going to be. The radius is so we&#x27;re just gonna call them are And we know that they&#x27;re the same because they both go from the center and touch this in conference. So then we can use the Pythagorean theorem. A squared plus B squared is equal to C squared and plug that and we know that the sea has to be the hypothesis. So that&#x27;s gonna be our four squared. And then So we&#x27;re gonna have our squared plus r squared. If we simplify this, we&#x27;re going to get to R Squared is equal to four squared and from here we can simplify the four squared into sixteen and divide both sides by two. That&#x27;s gonna give us that are squared, is equal to eight, and to isolate our all we have to do is take the square root of both sides giving us that r c go to the square root of eight. But the square root of eight we know eight is just two times two times two. We can take these out from under the square root, giving us that our is actually to swear it too. So c should be the answer

Heather Zimmers · Answer

Okay, I apologize for my wonky sketch, but hopefully it will get the job done. Our goal is to find the area of the semicircle E A B. So that would be half of the large circle. And we&#x27;re told that the radius of the small circle is three. So from C to D is three. So that tells us that the diameter of the small circle is six. And the diameter of the small circle is the radius of the big circle. And to get the area of the big circle, we would have pi r squared, which we now know is going to be pi times six squared. So that&#x27;s 36 pie. Remember, we just want the area of the semicircle, so half of it would be 18 pie, and that corresponds to answer choice D.

Problem

REWRITING A FORMULA Write a formula in terms of t…

Problem 34 Hard Difficulty

Answer

$\mathrm{C}=20 \pi$

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$\mathrm{C}=20 \pi$

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Lily A.

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Ron Larson, Laurie BoswellGeometry A Common Core Curriculum

Lily A.

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