A circle of unknown radius has a chord of length d whose midpoint is 1 foot from the circle. Find the radius, r, of the circle [in inches].
Please solve and give me compleate answer. I beliave there must be more than one answer please list all. THANKS!
2007-10-31 07:31:51 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
(d/2)^2 + (r-1)^2 = r^2, these three sides form a right triangle
Simplify,
(d/2)^2 - 2r + 1 = 0
r = (1/2)[(d/2)^2 + 1] ft, where d has the unit of ft
Therefore, once you know d, you can find a unique value of r.
If you want to represent r in inches, r times 12 gives you the answer.
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Ideas: 1 ft from the circle means r-1 ft from the center.
2007-10-31 07:38:54 · answer #1 · answered by sahsjing 7 · 0⤊ 0⤋
The chord touches the circle in two points (call them A and B). The radii from these points to the centre (C) will form an angle at the centre.
Drop a perpendicular onto the chord, from the centre. The length of this perpendicular (from the centre to the chord) is r - 12 inches (because it falls on the mid-point -- which we call D -- which is at one foot from the circle.
Take one of the two right-angle triangle, say ACD. We know that the hypothenuse is r (the value of the unknown radius) and that the side from C to D measures (r-12) in inches.
Therefore, the angle at the centre (angle ACD) can be found with:
Cosine(ACD) = r / (r-12)
The length of the chord can be found with:
AB = 2*AD = 2* SQRT[r^2 - (r-12)^2]
AB = 2*SQRT[24r - 144]
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In order to find an actual value for r, we need one more bit of info; either the length of the chord or an angle in one of the triangles we created. Having that, use the above equations -- backwards -- to solve for r.
If not, all you can say is that r must be at least 12 inches (then the chord is a diameter through the centre, placing its midpoint 12 inches from the circle)
2007-10-31 14:46:51 · answer #2 · answered by Raymond 7 · 0⤊ 0⤋
as i cant draw on this to help i will have to explain what i drew...
draw a circle with a random chord... draw in a line from centre to the midpoint of the chord and draw a line from the centre to where the chord touches the edge of the circle...
you should see a right angled triangle. it is a 1 by d/2 triangle so the third side is (1^2 + (d/2)^2)^0.5 = (1+(d^2)/4)^1/2
and you can see from the diagram this is also the radius!
2007-10-31 14:38:07 · answer #3 · answered by Anonymous · 0⤊ 0⤋
2 equations and 1 unknown. Without more info, you won't be able to find a unique solution.
r^2 = (12 in)^2 + (d/2)^2
2007-10-31 14:37:45 · answer #4 · answered by np_rt 4 · 0⤊ 0⤋