2001 AMC 8 Problems/Problem 9


Problem

To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid.

[asy] for (int a = 0; a < 7; ++a) { for (int b = 0; b < 8; ++b) { dot((a,b)); } }  draw((3,0)--(0,5)--(3,7)--(6,5)--cycle); [/asy] The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut off from the four corners?

$\text{(A)}\ 63 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 180 \qquad \text{(D)}\ 189 \qquad \text{(E)}\ 264$

Solution

The large grid has dimensions three times that of the small grid, so its dimensions are $3(6)\times3(7)$, or $18\times21$, so the area is $(18)(21)=378$. The area of the kite is half the product of its diagonals, and the diagonals are the dimensions of the rectangle, so the area of the kite is $\frac{(18)(21)}{2}=189$. Thus, the area of the remaining gold is $378-189=189, \boxed{\text{D}}$.

See Also

2001 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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