Question

Three circles are inscribed in a rectangle of width w and height h as shown. Two of the circles are congruent and are each tangent to two adjacent sides of the rectangle and to each other. The other circle is larger and is tangent to three sides of the rectangle and to the two smaller circles. What the ratio of h to w? Express your answer as a decimal to the nearest hundredth.

Answer 1

Answer: 0.69

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Explanation:

The two smaller circles have a height of h, so one circle has a height of h/2 = 0.5h

The radius of each smaller circle is (0.5h)/2 = 0.25h

Draw an xy axis. Place the bottom left corner of the rectangle at the origin (0,0)

The center of the lower smaller circle is at location (0.25h, 0.25h). Call this point A.

Let B be the center of the larger circle. It has coordinates (x,y). We don't know x, but we know that y = 0.5h since the center must be at the halfway point in terms of the height of this rectangle. So the larger circle has a radius of 0.5h

Draw a line segment connecting A and B. The length of this segment, call it d, is d = 0.5h + 0.25h = 0.75h. Note how I added the two radius values mentioned earlier.

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Summarizing everything so far, we have

A = (0.25h, 0.25h)

B = (x, 0.5h)

d = 0.75h

The distance formula is then used

d = distance from A to B

d = length of segment AB

d = sqrt( (x1-x2)^2 + (y1-y2)^2 )

0.75h = sqrt( (0.25h - x)^2 + (0.25h - 0.5h)^2 )

(0.75h)^2 = (0.25h - x)^2 + (-0.25h)^2

0.5625h^2 = 0.0625h^2 - 0.5hx + x^2 + 0.0625h^2

x^2 - 0.5hx + 0.4375h^2 = 0

From here you use the quadratic formula to get x = 0.9571067811865h approximately (the other solution is ignored as it's negative). See the attached image below if you're curious what the quadratic formula steps would look like.

This x value is the x coordinate of point B, which is the center of the larger circle. This spans the horizontal distance from the left edge of the rectangle to the center of the larger circle. The remaining horizontal distance is h/2 as it is the radius of the larger circle.

Therefore,

w = 0.9571067811865h + 0.5h

w = 1.4571067811865h

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We have turned w into a roughly equivalent expression that has an h in it, allowing us to find the ratio of h to w

h/w = h/(1.4571067811865h) = 1/1.4571067811865 = 0.68629150101527

When rounding to two decimal places, we get roughly 0.69