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Post by mds62 on Nov 21, 2020 9:27:23 GMT -5
Tbh this isn't too hard (some may even call it easy),
Here is the question
∆ABC is right angled at B. A semi circle whose center lies on BC , touches the sides AB and AC.
If the radius of the circle is 10cm, find the minimum possible are of Triangle ABC.
The first guy to solve this question can enslave me
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Post by Bagley on Nov 21, 2020 11:20:02 GMT -5
The triangle can have arbitrarily small area. Just let the center of the circle be B (which lies on BC) and let the edge of the circle pass through A (so the circle touches both AB and AC), then C can be as close to B as desired.
By "touch", did you mean that AB and AC must be tangent to the semicircle? That would make a bit more sense and make the problem slightly more difficult.
Furthermore, there's no reason you need to specify that it's a semicircle, you could just use a circle. That's because its only restriction is that it must pass through two points, so no matter where those two points occur on the circle, we can draw a semicircle that contains them. If it had to pass through three points, then it being a semicircle restricts your possible solutions.
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Post by mds62 on Nov 21, 2020 11:49:58 GMT -5
The triangle can have arbitrarily small area. Just let the center of the circle be B (which lies on BC) and let the edge of the circle pass through A (so the circle touches both AB and AC), then C can be as close to B as desired. By "touch", did you mean that AB and AC must be tangent to the semicircle? That would make a bit more sense and make the problem slightly more difficult. Furthermore, there's no reason you need to specify that it's a semicircle, you could just use a circle. That's because its only restriction is that it must pass through two points, so no matter where those two points occur on the circle, we can draw a semicircle that contains them. If it had to pass through three points, then it being a semicircle restricts your possible solutions. AB and AC are tangents to the circle/semicircle. I created the question myself. So forgive me if the question isn't worded properly |
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Post by Bagley on Nov 21, 2020 19:48:39 GMT -5
oh heck oh frick I forgot all about this, I'm tired but I'll try this tomorrow, I think the best way to solve it would be to use use y = sqrt(20x-x^2) as the semicircle with point B at (0,0) and find the area of the triangle for any point C along the x axis since that fixes A, and use the first derivative test, but I'll do that later ok bye
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