4 circles inscribed in a big circle of radius
rcm express the radius of the large circle in terms of r
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To find the radius of the large circle in terms of the radius of the small circle (r), let's analyze the scenario:
The small circles are inscribed in the large circle, which means they touch the large circle from the inside. If we connect the center of the large circle to the center of one of the small circles, we get a line segment passing through the center of the large circle. This line segment is also the radius of the large circle.
By drawing radii from the center of the large circle to the points where the small circles touch it, we can form four congruent right-angled triangles inside the large circle.
In each triangle, the hypotenuse is the radius of the large circle, and one of the legs is the radius of the small circle. The other leg is the distance from the center of the large circle to the point where the small circles touch it.
If we label this distance as d, we can apply the Pythagorean theorem to one of the triangles:
(r + d)² = (2r)²
Expanding and simplifying:
r² + 2rd + d² = 4r²
Combining like terms:
d² + 2rd - 3r² = 0
To solve this quadratic equation for d:
d = [-2r ± sqrt((2r)² - 4(1)(-3r²))] / (2(1))
Simplifying further:
d = [-2r ± sqrt(4r² + 12r²)] / 2
d = [-2r ± sqrt(16r²)] / 2
d = [-2r ± 4r] / 2
Now we can find the two possible values of d:
d₁ = (-2r + 4r) / 2 = 2r / 2 = r
d₂ = (-2r - 4r) / 2 = -6r / 2 = -3r
Since distance cannot be negative, we discard d₂ = -3r as a valid solution.
Therefore, the value of the distance between the center of the large circle and the point where the small circles touch it is d = r.
Finally, the radius of the large circle (R) is the sum of the radius of one of the small circles (r) and the distance (d):
R = r + d = r + r = 2r
Hence, the radius of the large circle is twice the radius of one of the small circles.
Therefore, the radius of the large circle (R) in terms of the radius of the small circle (r) is R = 2r.