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Question #52 on 2013 Inv. D:

Triangle ABC exists such that m[unparseable or potentially dangerous latex formula] ACB = [unparseable or potentially dangerous latex formula] and point P is the midpoint of segment AB. If CP = [unparseable or potentially dangerous latex formula] cm, BC = [unparseable or potentially dangerous latex formula] cm, and m[unparseable or potentially dangerous latex formula] CPB = [unparseable or potentially dangerous latex formula], find [unparseable or potentially dangerous latex formula]

Answer: [unparseable or potentially dangerous latex formula] cm

I've tried a few approaches with Law of Sines/Cosines but can't seem to figure it out. Anyone know how?

The formula for the circumradius of a triangle is [unparseable or potentially dangerous latex formula].

This is a right triangle so [unparseable or potentially dangerous latex formula].

also [unparseable or potentially dangerous latex formula].


Let me use [unparseable or potentially dangerous latex formula] as the side b.

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula].

[unparseable or potentially dangerous latex formula]

Using the Law of Sines for [unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

Substitute in [unparseable or potentially dangerous latex formula] and R into the thing:

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

Or if you don't want to be overly-complicated like I did on my last post, just use the law of cosines once!

You know that [unparseable or potentially dangerous latex formula] because the circum-diameter of a right triangle is the hypotheneuse, so the radius is the midpoint segment of it.

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

LAW OF COSINES!!!!!!!!!!!!!!!!!!!!!! MY FAVORITE!!!!!!!!!!!!!!!!!!!!!! :P :D :O :W

My method:

You know that [unparseable or potentially dangerous latex formula] is the geometric mean of [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula], and since [unparseable or potentially dangerous latex formula] we see that both must equal [unparseable or potentially dangerous latex formula]

That must mean [unparseable or potentially dangerous latex formula]

From there if you know your Pythagorean triples you can see that the the [unparseable or potentially dangerous latex formula]

Note that the since point P is a median, the area is cut in half, meaning the area of the smaller triangle is half of the entire area, or symbolically

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]