Two Simple Geometry Questions
On this day in 1809, Joseph Liouville was born. Also on this day in 1721, Johann Sebastian Bach published his famous Brandenburg Concertos which he dedicated to Christian Ludwig, Margrave of Brandenburg-Schwedt.
Today, I am posting two simple geometry questions. Since it's a weekend, feel free to take your time and enjoy them. Or, if you're feeling slightly more competitive, try to time yourself to see how quickly you can get the answers.
Question 1:
In the diagram above, given that F is the midpoint of CD and that 
, what fraction of square ABCD does 
occupy? Express your answer in lowest terms.
Question 2: In the diagram below, arc IFJ is one-quarter of a circle with center H. Given that HJ = IH = 6, and that EFGH is a rectangle with EH + HG = 8, what is the perimeter of the shaded region?
Were you looking for these answers parametrically?
Unless I'm missing something, which is possible, neither problem has a unique numerical answer.
For the first one, you can solve them algebraically.
In the second one, HF = EG because they're diagonals of the same rectangle.
lol, to the second one. That observation makes the problem trivial. Silly how I missed that, though I spent more time on the first one.
Algebra you say? Hmm, I wasn't using calculus. But then again, the first one has so many similar triangles running around, I probably got bogged down with all the info. But that's the great thing about geometry. There is sooo much interconnectedness. There is so much vitality.
I'll have to give that first one a second try when I get a chance.
For the first one, think about the slopes of BF and AE.