Constructing a TriangleDate: 09/29/2003 at 00:12:22 From: Bong Subject: constructing a triangle Let x be a given angle, and let and m and n be given lengths such that n > m. How can I construct triangle ABC such that AB = m, AC + CB = n, and the measure of angle ACB = x? I tried to add n and m and use the sum as the perimeter of the triangle but I can't construct a triangle with given segment AB and angle x. Date: 10/01/2003 at 03:23:00 From: Doctor Floor Subject: Re: constructing a triangle Hi, Bong, We will use several times that we can construct the circle containing the points that make a given angle y on a segment UV. This can be done in the following way: The circle we look for passes through U and V. We need a third point. From V draw the perpendicular line to UV. From U draw the line making an angle of 90-y with UV. These lines intersect in a point W. Angle UWV is then equal to y. So W lies on the required circle. That means that the circumcircle of UVW is the required circle. Since UVW is a right triangle, this circle has the midpoint of the hypotenuse UW as center. Now let's analyze the problem you have given. If we consider a triangle ABC with angle A = x, AB = m, BC = n1 and CA = n2, then we consider the point D on BC produced such that CD = n2, or BD = n1 + n2 = n. In the following diagram, triangle ACD is isosceles, and from angle ACD = 180 - x we see that angle ADB = x/2. Of course, with x given, we can construct an angle of x/2 by constructing the angle bisector. This gives the following construction: Start with AB of given length. Let C1 be the circle with center B and radius n. Let C2 be the circle of points making angle x/2 on AB. The points of intersection of C1 and C2 give possible D. C lies on BD. Construct the circle C3 of points making angle x on AB. The intersection apart from B of BD and C3 gives C. I wish to thank Antreas P. Hatzipolakis for suggesting to me this construction. If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ Date: 10/06/2003 at 03:22:38 From: Bong Alvarez Subject: Thank you (constructing a triangle) I wish to thank you for answering my question. It was a big help. I was wondering if this construction can be done using a straight edge and compass? Date: 10/06/2003 at 06:17:54 From: Doctor Floor Subject: Re: Thank you (constructing a triangle) Hi, Bong, Indeed it can. Perhaps you need some basic constructions, such as the ones given in the Dr. Math library at: http://mathforum.org/library/drmath/view/55076.html http://mathforum.org/library/drmath/view/54688.html If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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