Circle in Semicircle

Geometry Level 3

In the semicircle above, a small circle is tangential to the semicircle's diameter AB at point D, and tangential to the circumference of the semicircle at point C.

What is $\angle ACD$ ?

Depends on the location of C

45^ \circ

30^ \circ

60^ \circ

1 solution

Chung Kevin
Sep 30, 2016

Let O be the center of the small circle. Let AC intersect the circle again at E.
Then, the tangent at E is perpendicular to the diameter AB. (Do you know why?)

Hence, $\angle EOD = 180^\circ - 90^ \circ = 90 ^ \circ$ and thus $\angle ECD = \frac{90^\circ} {2} = 45 ^ \circ$ .

Because the dilation centered at C which maps the small circle to the big circle, maps E to A :p

Manuel Kahayon - 4 years, 6 months ago

Is it angle EOD or EFD?

Arunagiri Ravisankar - 4 years, 6 months ago

$O$ is the centre of the small circle.

Sharky Kesa - 4 years, 6 months ago

Ah, I forgot to define that.

Chung Kevin - 4 years, 6 months ago

Circle in Semicircle

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