2000 AMC 8 Problem 18
Attempt Problem 18 of the 2000 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
18.
Consider these two geoboard quadrilaterals. Which of the following statements is true?
The area of quadrilateral is more than the area of
quadrilateral
The area of quadrilateral is less than the area of
quadrilateral
The quadrilaterals have the same area and the same
perimeter.
The quadrilaterals have the same area, but the perimeter of
is more than the perimeter of
The quadrilaterals have the same area, but the
perimeter of is less than the perimeter of
Answer: E
Small hint:
Decompose both quadrilaterals into unit right triangles
Big hint:
Compute all four side lengths of each quadrilateral
Solution:
Assume that the pegs on this grid are separated by unit.
Note that region is a parallelogram with base and height making its area
We can split region into triangles, both with base and height This makes the sum of the areas This shows that both regions have the same area.
Note that each region has sides that are of length Region has unit sides, whereas region only has
The other side of region is clearly greater than which shows that region has the greater perimeter.
Thus, E is the correct answer.
Problem 18 in other years
- 1985 AMC 8
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