Question 1181824: Hi! I need a hint with how to make this construction: I was given three angles of a triangle and a segment R.
I need to construct a triangle with the given angles that is circumscribed about a circle with radius R.
I know that what we need to find is the circumcenter (by finding the perpendicular bisectors), but construction of parallel and perpendicular lines are not yet discussed so I cannot use that. Also, the term circumcenter was not yet discussed. The only constructions discussed so far were: duplicating segments, duplicating angles, and bisecting an angle.
Found 3 solutions by MathLover1, greenestamps, MathTherapy:
Answer by MathLover1(20850)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
(1) I will assume that you are talking about a classic construction using a compass and straightedge....
(2) The TRIANGLE is CIRCUMSCRIBED about the CIRCLE, so you want the INCENTER, not the CIRCUMCENTER.
The incenter is the intersection of the angle bisectors.
I haven't completely figured out how to do the construction. There might be an easy way to do it; but, as far as I can see, it is a very complicated one. I seriously doubt it can be done if the only constructions you have learned are duplicating segments, duplicating angles, and bisecting an angle.
Answer by MathTherapy(10553)  (Show Source):
You can put this solution on YOUR website! Hi! I need a hint with how to make this construction: I was given three angles of a triangle and a segment R.
I need to construct a triangle with the given angles that is circumscribed about a circle with radius R.
I know that what we need to find is the circumcenter (by finding the perpendicular bisectors), but construction of parallel and perpendicular lines are not yet discussed so I cannot use that. Also, the term circumcenter was not yet discussed. The only constructions discussed so far were: duplicating segments, duplicating angles, and bisecting an angle.
If you're referring to the INCENTER, then you have what you need to do the construction.
All you need to do is construct line segments that bisect each of the 3 angles. The 3 angle bisectors will be CONCURRENT at the INCENTER, which is also the center of the INSCRIBED circle.
With your compass, you then use the INCENTER (the inscribed circle's center) to draw the circle!
Also, the smaller segment of each angle bisector, that's located WITHIN the TRIANGLE, will be 3 of the RADII of the circle, if the triangle is an ACUTE one.
That's IT!!
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