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Math Help - Triangle construction problem - is it possible ?

  1. #1
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    Triangle construction problem - is it possible ?



    It's all, what i have. I know, for his parameters is 2 different triangles - but i don't now, how to counstruct it. If you know it, you are really, but really good mathematician. And.Thanks for help. It's not homework, nothing for school.
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Do you need to reproduce this drawing?

    First, draw AB with a ruler.
    Take your compass, draw an arc with length AB with centre A.
    Do the same with centre B.
    Where the arcs cut, it is the vertex C.
    Then, draw AC and BC with a ruler.
    Bisect any of the two sides of the triangle and join the lines of bisection.
    Where they meet indicate the centre of the triangle and at the same time the centre of the circle.
    Put your compass on the centre of the circle to point D, which is halfway between A and B, (or measure 4.8 units) and draw a circle.

    There you are.

    I hope that it's what you were looking for.
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  3. #3
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    Hm..Good..But i havent got length of AB ..And i can't calculate it, but i know, how to do it .
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  4. #4
    MHF Contributor Unknown008's Avatar
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    Well, I was assuming it was an equilateral triangle, because if I think about it, you can have various situations where AC and BC are 17, the circle inscribed in the triangle but various lengths of AB.

    Ok, let's get to calculations then.

    The let the angle FSE = theta.
    Then, angle FCE = 180 - theta

    First conclusion: CS + DS = 17 cos (\frac{\theta}{2})} ...1

    Elaborating CS + DS: CS + 4.8 = 17 cos (\frac{180- \theta}{2})

    Elaborating CS: CS = \frac{4.8}{cos(\frac{\theta}{2})}

    Using what we just found to get to the first conclusion (ie replace the values of CS and DS into 1);

    \frac{4.8}{cos(\frac{\theta}{2})} + 4.8 = 17 cos (\frac{180- \theta}{2})

    4.8 + 4.8cos(\frac{\theta}{2}) = 17 cos (\frac{180- \theta}{2})cos(\frac{\theta}{2})

    4.8 + 4.8cos(\frac{\theta}{2}) = 17 cos (\frac{180- \theta}{2})cos(\frac{\theta}{2})

    But now, cos(180-theta) = sin(theta)

    So,
    4.8 + 4.8cos(\frac{\theta}{2}) = 17 sin (\frac{\theta}{2})cos(\frac{\theta}{2})

    4.8 = 17 sin (\frac{\theta}{2})cos(\frac{\theta}{2})- 4.8cos(\frac{\theta}{2})

    4.8 = 17 cos (\frac{\theta}{2})(sin(\frac{\theta}{2})-4.8)

    Well, I'll use wolframalpha to find that angle.

    http://www.wolframalpha.com/input/?i=4.8+%3D+17+cos+(\frac{x}{2})(sin(\frac{x}{2})-4.8)

    So, theta = 3.14 rad, which is 174.4 degrees.

    So, angle FCE = 5.58 degrees.

    From here, you can draw AC first, draw the angle, draw BC, then join AB, and then to find the centre for the circle, find the intersection of the lines DC and AE or any other pair. Open your compass at 4.8 units and draw your circle.

    I hope that's it.
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  5. #5
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    Dear bakyx,
    You show a circle inscribed in an equilateral triangle 17 units on a side and a radius of circle of 4.8 units. Numbers do not compute. Do you know the properties of 30-60-90 triangles.If not look it up and straighten out dimensions.
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  6. #6
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    Quote Originally Posted by BakyX View Post
    It's all, what i have. I know, for his parameters is 2 different triangles - but i don't now, how to counstruct it. If you know it, you are really, but really good mathematician. And.Thanks for help. It's not homework, nothing for school.
    The bad news first: I can't show you how to construct the triangle without doing some calculations.

    1. The triangle in question is an isosceles triangle with the base b, the area a and the height (2r + x). Compare the attached sketch.

    2. The area of the triangle is calculated by:

    a = \frac12 \cdot b \cdot (2r + x)

    Divide the triangle into 3 triangles with the common vertex S. Then the heights of the 3 triangles are the radius r = 4.8. Then the area of the complete triangle is calculated by:

    a = 2 \cdot \frac12 \cdot 17 \cdot r + \frac12 \cdot b \cdot r

    3. The indicated triangle is a right triangle. Use Pythagorean theorem:

    (2r + x)^2+\left(\frac12 b\right)^2=17^2

    4. Solve the system of equations for (a, b, x):

    \left|\begin{array}{rcl}\frac12 b (9.6+x)&=&a \\ 81.6+2.4 b &=& a \\ (9.6+x)^2+\frac14 b^2 &=& 17^2 \end{array}\right.

    5. This system has 4 solutions. Neglecting the solutions with a = 0 or negative side length you'll get 2 plausible solutions:

    (a,b,x)=\left(120, 16, \frac{27}5  \right) ........ or

    (a,b,x)=\left(\frac{36}{25} \sqrt{769} + \frac{516}5, \frac{3}{5} \sqrt{769} + 9 ,  \frac{ \sqrt{769}}2 - \frac{123}{10} \right)
    Attached Thumbnails Attached Thumbnails Triangle construction problem - is it possible ?-dreieck_um_kreis.png  
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