Question 251467: construct an isosceles triangle with base BC=5cm and the vertical angle BAC=30 degree.(do not construct by calculating the base angles).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! build a rough sketch of your isosceles triangle with base = 5 cm.
label the triangle ABC with the base equal to BC.
Drop a perpendicular from A to intersect with BC.
Since the triangle is isosceles, this perpendicular will also divide the angle into 2 equal parts, and will also divide BC into 2 equal parts.
You have 2 right triangles.
Assuming the altitude intersects BC at D, the right triangles are:
ABD and ACD
the 30 degree angle will be split in 2 equal parts, each of which is 15 degrees.
the base of 5 cm will be split in 2 equal parts, each of which is 2.5 cm.
sine of 15 degrees = opposite / hypotenuse = 2.5 / x
this makes x = 2.5 / sin(15)
this makes x = 9.659258263
x is equal to one of the equal legs of your isosceles triangle.
your isosceles triangle dimensions are 2 legs of 9.659258263 and one leg of 5.
now that you have one of the sides of your triangle, you can construct your isosceles triangle given one side and the base.
the following link shows you how to do that.
http://www.mathopenref.com/constisosceles.html
I was not able to find how to construct given the base and the vertex angle directly.
you can, however, construct a 30 degree angles using the method also described at this website.
once you do that, you can then construct a line that bisects that angle.
once you do that you can then construct your base of 5cm and slide it up and down that angle bisector while it is perpendicular to it until you touch both sides of the 30 degree angle which should then give you the two equal legs of that triangle.
if you scroll down the website, it shows you how to construct a 30 degree angle.
it also shows you how to construct an angle bisector.
the procedure is visualized in the following link.
http://theo.x10hosting.com/problems/251467.html
the procedure is:
1. construct a 30 degree angle.
2. construct an angle bisector to that 30 degree angle.
3. construct a line perpendicular to that angle bisector and measuring 5cm total with 2.5cm on each side of the angle bisector.
4. slide that perpendicular line up or down the angle bisector until the ends of it touch the sides of the 30 degree angle.
you should get a perfect fit with the sides of the 30 degree angle intersecting with the base at a length of 9.659... cm each.
I've never done this before, but I don't see why it wouldn't work. If the angle is 30 degrees, and the base is 5cm, it has to touch the sides of the 30 degree angle at the correct points.
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