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Washington State Math Olympiad
Hints and Solutions
2015 Grade 8 Geometry
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1) In the diagram shown, what is the angle of VAB?
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- Angle ACB is vertical with angle YCZ, so it is 30 degrees.
- Angle ABC is vertical with angle WBX, so it is 80 degrees.
- Angle BAC is 180 - 30 - 80 = 70 degrees
- Angle VAB is supplementary with angle BAC and is therefore 180 - 79 = 110 degrees
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2) A data set consists of seven different positive whole numbers. The mean of the data is 76. What is the largest possible median of the data?
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- For the median to be maximum, the 3 numbers below it must be as small as possible = 1,2 and 3
- The 7 members of the data set must sum to 76x7 = 532, so the median and the 3 numbers above it must sum to 532 - 6 = 526
- For the median to be maximum, the median and the 3 numbers above it must be as high as possible, so if they are all different they must be 130, 131, 132 and 133, so the highest median is 130
- As a check:
(1+2+3+130+131+132+133)/7 = 532/7 = 76
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3) Rotate triangle XYZ 90 degrees clockwise (↷) about point Y. What will the translated coordinates of point Z be?)
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- Point Y is (5,6)
- Translate the entire triangle to make point Y the origin:
Z = (3-5,1-6) = (-2,-5)
- To rotate a point (a,b) clockwise 90 degrees, it's new coordinates become (b,-a)
- Translated, rotated Z point = (-5,2)
- Translate back into original coordinates:
Original Z rotated 90 degrees clockwise = (-5+5,2+6)= (0,8)
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