Problem 18 · 2014 Math Kangaroo
Stretch
Geometry & Measurement
spatial-reasoning
The sides of a rectangle are 6 cm and 11 cm long. You select one of the long sides. Then the angle bisectors of the angles at the two ends of this side are drawn. They split the opposite long side into three pieces. How long are these pieces?
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Answer: E — 5 cm, 1 cm, 5 cm
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Hint 1 of 2
An angle bisector of a 90° corner makes a 45° line, which drops at 45° to the opposite side.
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Hint 2 of 2
A 45° line travels the same distance sideways as the rectangle's short side, 6 cm.
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Approach: use the 45° bisectors to locate the two split points
- Each corner of the chosen long side is 90°, so its bisector is a 45° line crossing the 6 cm gap to the opposite side.
- A 45° line moves 6 cm sideways while crossing, so the left bisector meets the opposite side 6 cm from the left end, and the right bisector meets it 6 cm from the right end (5 cm from the left).
- The two marks at 5 cm and 6 cm split the 11 cm side into pieces of 5 cm, 1 cm and 5 cm.
- So the pieces are 5 cm, 1 cm, 5 cm.
Mark:
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