Saturday, May 24, 2008

Geometric Construction and the Sinus Function

Today I have a fantastic question from Yahoo Answers about geometric construction. The idea of geometric construction is to use a compass and a ruler (with no measurements) to construct different geometric shapes and figures.

Notice: to answer this question you need to know basic constructions: copying a segment, creating a segment X times larger than another, find a perpendicular bisector, and copy an angle. Without those you'll be lost.

Here is the question:

A triangle has sides a, b, and c. The ratio a/b = 7/4. You are given side c and the radius of the circumcircle, r. Construct the triangle.

This may seem simple, but it's a little more sophisticated than that. To construct that triangle, we need the law of sines.

The law of sines says that in a triangle, a/sin A = b/sin B = c/sin C = 2R (side a and angle A are opposite, R is the radius of the circumcircle). Let's play with this law a little:
a/sin A = b/sin B
That means:
a/b = sin A / sin B

In our triangle, a/b = 7/4. So all we need to do is find two angle whose ratio of sines is 7/4. So how do we do that?

The definition of sine A is the y-coordinate on the unit circle with the angle measure of A (read here about the unit circle). First, construct two segments, a' and b' in a way that a'/b' = 7/4. Now create a circle with a radius of more than a' and two perpendicular axes that intersect at the center of the circle. Position a' in the circle in such a way that it's perpendicular to the x-axis and touches the circle in one point (yet not tangent to it). Construct the line from the origin to the point of intersection of a' and the circle. Call the angle between the x-axis and that line angle A. Do the same thing for b' and angle B.

Since we used the definition of sine, we now have two angles with a sine ratio of 7/4. Now we need to create the triangle we want.

First, construct segment c. Since the circumcenter, the center of the circumcenter, is on the point of intersection of all perpendicular bisectors, construct the perpendicular bisector of segment c. Now, since the radius is given, use the end point of segment c and the perpendicular bisector to find the circumcenter and draw the circumcircle.

Now, when segment c is inside the circle, copy angle A to one of its sides and angle B to the other side. Complete the triangle, and you are done.

Feel free to send in more questions!



Anonymous said...

It's so easy to choose high quality [url=]replica watches[/url] online: [url=]Rolex replica[/url], [url=]Breitling replica[/url], Chanel replica or any other watch from the widest variety of models and brands.

Anonymous said...

Hello. In a crisis, fell revenue from sales [url=][/url] . Tell my what can be done. Thanks in advance.
Vsem privet. V uslovijah krizisa upal dohod ot prodazh [url=][/url] . Podskazhite chto mozhno sdelat'. Zaranee Spasibo.

Anonymous said...

Now is a great time to trade currencies with the world economic problems. People are cashing in by trading forex right now, the world money sistuation is a mess so why not make profit off it? Keep your money offshore where it is safe!

A good Forex broker is 1pipfix, 1pip spreads and the best top rated of forex brokers with metatrader 4

1pipfix is a partner broker of

Anonymous said...

Best Forex Brokers stp ndd ecn
free metatrader demo, the top forex broker is 4runnerforex tight low spreads.

Anonymous said...

[url=]Poker[/url] is going to be fully licenced and legit in the USA by the end of 2012. This change will allow more than 20 million potential players to play poker daily

I believe the best source for [url=]online poker[/url] information, updated and bonuses.


Anonymous said...

[url=]supra trainers[/url]

Anonymous said...

[url=][b]outlet ugg[/b][/url]
[url=]Chestnut Ugg Boots[/url] nib8wot
[url=]Gold Ugg Boots[/url] eid7rnm
[url=]bailey button ugg boots[/url] mvp2pwk
[url=]bailey button ugg boots[/url] gno0cqd

Anonymous said...

[b][url=]ugg boots uk[/url][/b] When freight brokers or logistics products and services providers arrange LTL routes, they typically look at the routes for being long-term. The benefits into the buyer are plainly from the price savings for the shipper who are able to then retain costs lessen. The gain into the carrier is the fact that the routes may be set up for many years assuring a dependable income from the shippers..

[b][url=]louis vuitton purses[/url][/b] Sometimes,Discount Coach Purses you may observe an actual broadly identified get the top rated part dressed in Herve Leger strapless check out an honor get jointly on Television set, at this time, it is actually ordinary, you think about. If you have any worries about your own overall health or maybe the well being of one's child, you need to normally consult with having a medical doctor or any other health care professional. Remember to evaluation the Privacy Plan and Phrases of Use ahead of using the site.

[b][url=]ugg boots outlet[/url][/b] The DAX futures agreement is one of the couple of Eurex futures out there for trade from the America. The popularity from the DAX futures agreement for American buyers is easily comprehended. The bear sector in between 2000-2002 made increased regulation for working day traders within the United states of america as many were unprepared for any down sector.

[b][url=]louis vuitton official website[/url][/b] To the previous pair of many years, wi-fi Bluetooth headsets have redefined what cellphone customers thought they understood about compactness and fashion. These neat equipment are frequently modest enough to fit above one ear, and their designs get a lot more smooth and present day through the day. Whilst most Bluetooth wireless headsets appear in this over-the-ear model, you'll be able to uncover other shapes too, like a lot more regular headsets with bands that go at the rear of or above the head.

[b][url=]louis vuitton bags[/url][/b] When you are one among people that find to have the true assortment of designer purses, then you definately can't do it effectively with no knowing your Chanel. Not surprisingly, becoming in a position to find out authentic Chanel purses requires you either to review a little bit of those wonderful brands' history or be the expert as you acquire one bag to a different. A single vintage magnificence that you just really have to know inside out is definitely the 255, a Chanel classic that has reigned for a lot of decades now..

Anonymous said...

cheap ugg boots There are several persons who really like to obtain bags which can assist them a whole lot. Should you are going to acquire your affordable purse on line make sure you are very careful. You also should be confident that you have gotten the original rather than just the reproduction purses. For those who like investments, make investments within a genuine handbag. They are going to previous more time and may keep up a lot better than a replica. More than time you will note a better value that has a genuine Mentor handbag. demands may well not hold sway in Ramallah any much more than it does in Cairo, Tunis or another capital of a rapidly-changing Center East. Safety Council resolution condemning Israeli design of settlements on occupied territory, in spite of concerted strain from the Obama Administration to stand down. isolated as the Council's only no vote (it can be nay counts being a veto, of course) with a resolution that many noticed as consistent with the Administration's individual stance on Israel's settlements.

louis vuitton sale What's really worth mentioning is the fact that the check out is water-resistant to a hundred meters. This check out can be obtained in a cost of $2670.00. It appears a little bit expensive, but taken the stylish style plus the very long assurance into account, you won't regret to have acquired this kind of an excellent look at..

uggs outlet With these handbags' brand-new that means to ladies and women alike, one can only be chuffed for your amazing choice of these equipment. There are bags for organization functions, and you can find these night bags to get a passionate dinner. Please read on for more about photos of sanitary napkin disposal bags and pvc bike sissybar bags..

uggs outlet Research the problem after which clear up it. It may be as easy as a lacking element or a thing as challenging being an environmental boo-boo. Regardless, be for the scene along with your attempt to unravel hat on and head to it.. Eurohandbag sells louis vuitton nomade leather-based. Additionally, it supplies purses, products, wallets, and purses from prime designers this kind of as Fendi and Orlando Dior, to call a handful of. To be able to get a bag with the internet site, you could select from exquisite colors these types of as reddish, black, shiny, and brownish.

uggs boots clearance In addition, if you will find any special specifications pertaining for the cargo, these types of as refrigeration, the broker is definitely the party that usually finds the top provider for your career. Brokers should be properly trained and certified just before they might perform company. They have got to know about premiums, distances, insurance protection, and a lot more.

Anonymous said...

Moncler Outlet a large number of The Su Hongye in Chongqing something, cannot return in,[url=]moncler outlet[/url] wrote that all the daughter decide, as long as she said [url=]moncler outlet[/url] the study. the foreign flavor of the new people are very against [url=]moncler coats[/url] the old pick auspicious day for marriage, advocates pick ocean life. that the most unfavorable marriage Gregorian calendar in May, the Gregorian calendar in June is better marriage, but they are already engaged in June, the so extended to early September wedding. It is stated too much focus on date,Monday 23 is a great day for marriage, especially on Wednesday; 4,561 days just like a bad day, the result is that they pick on Wednesday smiles: should have been that guy to come out of Yuen Long Cao tricks. Mei smiles: short, you hate the European students, pattern names up. Chosen to get married that Moncler Outlet Wednesday, the weather is like summer, hot interest. The way in which I needed, Jiao days Fortunately, today I did not do groom. The church was air-conditioned, Ts wearing a black wool dress, too busy sweating, I think he white collar having a ring, to obtain another yellow sweat soaked and soft. I afraid the entire of his plump body In Khan, how a candle into a pool of oil. Miss Su is also tight ugly. line up at the wedding, bride and groom smiling face, no expression of Ku Buchu, all unlike the dry wedding, but rather no, this is not on the scaffold, is a, is, like Moncler Boots a public host to pickpockets signs with punishing those hardened criminals in the face. I occur to believe that I own wedding ceremony, under those thousands of Kui Kui like to be inevitable cracked pickpockets. That made me realize the kind of joyful, smiling faces of happy wedding pictures were never taken to. find! Great find! I am thinking about is to see how you prefer her the same day. avoided her not to see, just say a few words with Miss Tang - chien hop heavy heart that, like truck unloading the parcel the next smash gravitropic only strange Xinmei won't hear - would be a bridesmaid the day, saw me and Moncler Scarf inquired about not to fight, said the ceremony complete line, we sprinkle colored paper Fou new body, when, in support of I will not have hands, afraid which i take the chance to throw hand grenades, nitrate sulfuric acid spill. She asked me later on plans, I informed her to go Sanlv University. I think she may not wish to hear your company name, so I didn't mention a thing you.

more information you can go to

[url=]moncler london[/url]
[url=]moncler women[/url]
[url=]outlet moncler[/url]
[url=]moncler gamme bleu[/url]
[url=]moncler down[/url]
[url=]moncler parka[/url]
[url=]moncler branson[/url]
[url=]moncler outlet usa[/url]
[url=]moncler grenoble[/url]
[url=]moncler outlet[/url]
[url=]moncler jacket[/url]

Anonymous said...

top [url=]free casino bonus[/url] hinder the latest [url=][/url] unshackled no consign hand-out at the chief [url=]casino

Anonymous said...

Moncler Himalaya Ebay Moncler Himalaya Ebay , hD3xxTwTTl
Moncler Himalaya Rosso Moncler Himalaya Rosso , FwND6yAx4dW
Moncler Himalaya Prezzo Moncler Himalaya Prezzo , qJwlL3gQDz
Orig Moncler Daunenjacke Himalaya Orig Moncler Daunenjacke Himalaya , PkK0IqAviR
Moncler Himalaya Nero Lucido Moncler Himalaya Nero Lucido , zUhbKfTMm
Branson Moncler Jacket Branson Moncler Jacket , TwHItSukP
Moncler Branson It Moncler Branson It , hCzh9VtOGo
Moncler Branson Ha La Cerniera Moncler Branson Ha La Cerniera , EpBOaHboG.
Branson Moncler Grigio Branson Moncler Grigio , JvVBrEieD
Moncler Branson Falso Moncler Branson Falso , lWey8BzFYl
Moncler Branson Ebay Moncler Branson Ebay , KeLQyZqxY.

Anonymous said...

Doudoune Moncler Quincy Homme Doudoune Moncler Quincy Homme , xL9pvMfUTf
Doudoune Moncler Pop Star Doudoune Moncler Pop Star , PwEC6qRn9nF
Doudoune Moncler Olympia Doudoune Moncler Olympia , rRxcY5oIPx
Doudoune Moncler Nice Doudoune Moncler Nice , TrN3CyQqvN
Doudoune Moncler Mod猫le Branson Doudoune Moncler Mod猫le Branson , xMkdVoKHu
Doudoune Moncler Luxembourg Doudoune Moncler Luxembourg , HnUWbJgyV
Doudoune Moncler Karak Doudoune Moncler Karak , vMyi4HxLWn
Doudoune Moncler Italy Doudoune Moncler Italy , MvHHiYdxJ.
Doudoune Moncler Femme Fourrure Doudoune Moncler Femme Fourrure , DtYIqAsdG
Doudoune Moncler France Doudoune Moncler France , sUyw5RbGGw
Doudoune Moncler Ebay Doudoune Moncler Ebay , HuVExHlwZ.

Anonymous said...

4nDsk cheap ghd
aAfl michael kors handbags outlet
dDjc ghd nz sale
7tNgo ugg sale
9rTrs ghd sale uk

Anonymous said...

fTag coach outlet
yJde cheap ugg boots
xPrb michael kors
8jKvv ugg boots cheap
4qQek chi straighteners
5iPpi michael kors handbags
9pCka nike nfl jerseys
9vNco ghd nz sale
2yNxu north face outlet
8kFni botas ugg españa
1tQbl mini ghd
7cFoc michael kors purse
0kLdx nfl jerseys
3oLqy ghd planchas
8zHqe ugg boots sale

Anonymous said...

JdkIha RqeAbo [url=]シャネル バッグ 中古[/url] VuuYug MgwFhz [url=]コーチファクトリー セール[/url] YykMic QwzMgn [url=]クリスチャンディオール バッグ[/url] HduWgw XsaFnu [url=]グッチ バッグ バンブー[/url] KssRkd HtmDrt [url=]ロンシャン バッグ ケイトモス[/url] VqbDsh DrwBua [url=]ヴィトン キーケース 定価[/url] FkcDdt XyxYsz [url=]プラダ トート デニム[/url] ZhfBju TciVkg [url=]アグ ブーツ アウトレット[/url]

Anonymous said...

RraTfs LgbQvz [url=]クロエ アウトレット[/url] NwkFrl BskBef [url=]クリスチャンルブタン ブーツ[/url] GjxIyl GxcVyi [url=]MCM 財布[/url] ZtyFjx TuvKnd [url=]ルイヴィトン アウトレット[/url] HzdUjx SdtKil [url=]クロエ 財布[/url] HdjTti GwpBqs [url=]ルブタン パンプス[/url] EanXxw KywYny [url=]グッチ[/url] PjgQpd BtrPfr [url=]TUMI スーツケース[/url]