Problem 5 · 2012 AMC 8
Easy
Geometry & Measurement
match-total-heights
In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is X, in centimeters?
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Answer: E — X = 5.
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Hint 1 of 2
Don't try to find every coordinate. In any all-right-angle figure, the vertical segments going up must total the same as the segments going down — otherwise the outline wouldn't close back to where it started.
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Hint 2 of 2
This is the closed-loop balance for staircase figures: sum of upward steps = sum of downward steps (and likewise left = right). Add up the verticals on each path and set them equal.
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Approach: vertical segments up = vertical segments down (closed loop)
- Walk the outline: every bit you climb up has to be undone coming down, since the figure closes. So total "up" height = total "down" height.
- The verticals on one side add to 1 + 2 + 1 + 6 = 10.
- The verticals on the other side add to 1 + 1 + 1 + 2 + X = 5 + X.
- Set them equal: 5 + X = 10 ⇒ X = 5.
- You'll reuse this on every "find the missing length in a right-angled figure": the up-segments and down-segments balance, and so do left and right — no need to chase individual corners.
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