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#1 2007-06-09 16:41:07

Tri
Member
Registered: 2007-06-09
Posts: 3

Triangle inside a circle.

Hi im a yr 9 student and im needing some help.
Im wondering if a equilateral triangle is enclosed in a circle with 100m radius how to find out the area.  So far i arrived at 14907.11985cm squared. How i did that was divide the triangle up into three iscosels triangles.  f_iscolestriam_1e16d47.gif
Then the angles of each where 30 degrees,30 degrees and 120 degrees. I then plussed the two equal sides which equaled 200 cm and times them by 120/180 to find the base of 133.33333333 recurring. i then used this site to find the area: http://www.ajdesigner.com/phptriangle/isosceles_triangle_area_k.php

this is based on all my dodgy guesswork so i was wondering if you could help me dizzy

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#2 2007-06-09 17:28:46

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,968

Re: Triangle inside a circle.

The cosine formula gives the third side when two sides and the angles are known.
a²=b²+c²-2bcCosa
In this case, it is rightly identified that angle a is 120 degrees.
Therefore, a²=2r²-2r²Cos120. Since cos 120 is -1/2, The area can be computed using the formula. Afther the longest side of a triangle is known, the formula for area of a triangle (in this case, an equilateral triangle) whose 3 sides are known can be calculated using the Hero's formula.
Area=√(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 and a,b,c are the three sides. When the triangle is equilateral, the area is √3/4(a²).
In this case, a²=3r², area=3√3r²/4.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2007-06-09 18:20:29

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Triangle inside a circle.

This is how I did the problem:
Let the side lengths of the equilateral triangle be x.

Given: AD = AB = BD = x

Construct: Angle bisector of A
                Line from E to D

Consider ΔADE and ΔACD.
              /_ADE = /_ACD (right angles)
              /_DAE = /_CAD (common)
              ΔADE ~ ΔACD (AA)

Hence,

Also,

Hence,






Hence,

That's 3247.6m²

Am i right?

Last edited by Identity (2007-06-09 20:25:08)

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#4 2007-06-09 19:29:47

Tri
Member
Registered: 2007-06-09
Posts: 3

Re: Triangle inside a circle.

I am very confused.
Any chance you could explain all this in simpler terms I havn't learnt like anything your saying and dont really understand it.

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#5 2007-06-10 02:06:02

shocamefromebay
Member
Registered: 2007-05-30
Posts: 103

Re: Triangle inside a circle.

f_iscolestriam_1e16d47.gif
i divided the triangles again
now we have 6 rite triangles taht are congruent to each other
its a special triangle
so its like
30° side = x
60° side = x(3^.5)
90° side = 2x

and from this we no that the side opposite the 90°angle is 2x
and taht is also the radius
and the radius is 100 m
so from taht we no taht x = 50 m
so taht means teh side opposite the 60° angle is 50(3^.5)
and teh side opposite the 30° angle is 50 m
so i we multiply taht by 6
then we have the perimeter of the triangle
which is 300 m

and we just use the area of an equilateral triangle to find its area
and thats it

pretty simple rite?

Last edited by shocamefromebay (2007-06-10 02:32:14)

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