Two Geometry Questions - If you DARE!

[tbroker]

1. A quarter ($.25) is 3/4 inch in diameter and when it is 7 feet from the eye it blocks out the moon which has a diameter of 2160 miles, how far is the moon from the earth (show your work so you are not just googling it)

2. In the Triangle ABC, D and E are midpoints of AC and BC. The segments AE and BD intersect at F. How can you prove that region a2 and a4 have equal areas.

[zeira]

Spoiler:

For (2) you know [BDC]=[BDA] and a1=a3. If a1=a3 is not obvious, draw DE and you have [DAB]=[EAB].

[kartelite]

1) 7/(.75/(12*5280*2160)) = 1 277 337 600 feet

2) Area ABE = Area ACE, so a2+a3=a1+a4. Area ABD = area BDC, so a1+a2=a3+a4.

So a2=a1+a4-a3, a1+a2=2*a1+a4-a3 = a3+a4, thus 2*a1=2*a4, so a1=a3. Therefore, a2+a1=a2+a3=a1+a4, or a2=a4.

[i_love_u]

Just to quickly add on to kartelite's work, part 1 is done using trigonometry. I cant draw the diagram here, but it is tan theta = 3/4 inch/7 feet. tan theta is also equal to 2160 miles/ x where x is the distance between the person holding the coin and moon. Equate the two RHS (obviously tan theta = tan theta = LHS) and u will get the answer of kartelite.