Bash or No Bash?
Square
A
B
C
D
has 3 quarter circles drawn on as shown above.
Points E and F are the intersections of the arcs. Point G is the midpoint of arc E F . Find the value of ∠ E G F in degrees.
The answer is 165.
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2 solutions
Nice solution. +1. It's not really bashy. It's quite elegant actually. Bashing it would be like using coordinate geometry to get the answer, which is really ugly to even try.
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Which is my usual method, Sharky. You know, when you're working on a job and you have to have accurate results ASAP and do it reliably, you bash. But I figured you were up to something.
How do you know the inscribed polygon has 24 regular sides?
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1) Note that BCF is equilateral
2) Thus angle FBA = 30
3) Thus A, E, C are vertices of a 12 sided polygon
4) G is equidistant from E, F
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how did you know that BCF is equilateral?
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@Francis Torres – How does one construct an equilateral triangle using a compass and a ruler? There's your answer.
∠ C B E = ∠ A B F = 3 0 o and ∠ E B G = ∠ G B F = 1 5 o . As E B G and G B F are congruent isosceles triangles ∠ E G B = ∠ F G B = 8 2 , 5 o so ∠ E G F = 1 6 5 o
How do you know that ∠ C B E = 6 0 ∘ ? For me it seems more likely it's 3 0 ∘
How do you know that ∠ E B G = 8 2 . 5 degrees?
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It was a mistake, it's ∠ E G B . because I said before that ∠ E B G = 1 5 o
João Arruda is right, ∠ C B E = 3 0 ∘ .
Nice! For symmetry reasons, points E , G , F are vertices of a 2 4 sided polygon. Thus, we solve the equation
( 1 8 0 − x ) 2 4 = 3 6 0
to find angle x = 1 6 5
Is this bashy?