Cornered baby circle

Geometry Level 4

The above shows two circles internally tangent to each other at one point.

The smaller circle is inscribed inside the region between the 2 straight lines $AB,BC$ and the larger circle. Also $AB \perp BC$ and $|AB| = |BC|$ .

If the radius of the larger circle is $5\sqrt2$ , find the radius of the smaller circle to 3 decimal places.

The answer is 5.858.

2 solutions

Ajay Sambhriya
Jan 23, 2017

R * (rt 2 +1) /rt2 =10 R * (rt2 + 1) = 10rt2 R = 10 rt2 / ( rt2 + 1 )

Nice question. To make sure that the answer is unique, it should probably be mentioned that $|AB| = |AC|$ and that $AB \perp AC$ . Most people will make these assumptions but it may be better to be explicit.

Brian Charlesworth - 4 years, 4 months ago

i think i mentioned that AB is perpendicular to BC....but they removed i think when they updated it .

Ajay Sambhriya - 4 years, 4 months ago

Ah, o.k.. They might have removed the condition that $AB$ and $AC$ are the same length as well, but that is still required for the answer to be unique.

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth – I have updated it see now

Ajay Sambhriya - 4 years, 4 months ago

Thanks for bringing this into our attention. We have accidentally edited the problem wrongly.

Brilliant Mathematics Staff - 4 years, 4 months ago

Marta Reece
Apr 16, 2017

$AB=BD=5\sqrt{2}$

$DC=CE=R$

$AC=R\sqrt{2}$

$AD=2\times AB=2\times 5\sqrt{2}=10\sqrt{2}$

$AD=R\times \sqrt{2}+R$

$R(1+\sqrt{2})=10\sqrt{2}$

$R=\frac{10\sqrt{2}}{1+\sqrt{2}}\approx5.858$

Cornered baby circle

The answer is 5.858.

2 solutions

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