A pentomino is made up of five squares where two squares are joined along an entire side. Two pen- tominos are the same if they
can be matched by turning or flipping. How many different pentomino shapes are possible?
1 answer:
There are 12 such shapes. One can start with two squares, adding squares around the edge and rejecting shapes that have been previously seen. Repeating until you have 5 squares results in 12 distinct shapes.
_____
2 shapes are possible with 3 squares: L, I
5 shapes are possible with 4 squares: L, I, N, T, O
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Answer:
27 degrees
Step-by-step explanation:
The easy way is by remembering the formula (a-b)/2=c, where a is the larger angle, and b is the smallest angle. (90-36)/2=27.
The longer, more drawn out answer goes as follows. See the image to understand the notation I use:
- AOE + BOD + BOA + DOE = 360
- AOE + BOD = 90 + 36 = 126
- BOA + DOE = 360 - AOE - BOD = 234
- Since the sum of a triangle's angles is 180, ODE = (180 - DOE) / 2
- Likewise, OBA = (180 - BOA) / 2
- Since CDE is 180, CDO = 180 - ODE = 180 - (180 - DOE) / 2
- Likewise, CBA is 180, so CBO = 180 - OBA = 180 - (180 - BOA) / 2
- The interior angles of the irregular polygon CBOD add up to 360, so CBO + CDO + BOD + BCD = 360.
- Substituting what we already found, 180 - (180 - BOA)/2 + 180 - (180 - DOE)/2 + 36 + BCD = 360
- Cleaning it all up, we get 180 + (BOA + DOE)/2 + 36 + BCD = 360
- As we found in line 3, BOA + DOE = 234, so substituting that in, 180 + 117 + 36 + BCD = 360
- Finally, solving for BCD (360 - 36 - 117 - 180) we get our answer, 27
Note: The long drawn out method shown above is a way to derive the formula for the secant theorem. You do not need to use this method every time. Just remember, large angle minus small angle, all divided by 2. That is it.
