AMC 8 1996 Problem 22
AMC8 1996 年第 22 题
三角形★★★☆☆
The horizontal and vertical distances between adjacent points equal 1 unit. What is the area of triangle ?
相邻点之间的水平和垂直距离均为 1 个单位。三角形 的面积是多少？
- A.
- B.
- C.
- D.1
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
takes up half of the 4x3 grid, so it has area of 6.
has height of BF = 2 and a base of AD = 4, for an area of .
has height of BE = 1 and a base of CD = 3, for an area of
Note that can be found by taking , and subtracting off and .
Thus, the area of , and the answer is .
There are other equivalent ways of dissecting the figure; right triangles and rectangle can also be used. You can also use and trapezoid .
Using the Shoelace Theorem, and labelling the points A(0,0), B(3,2), C(4,3), we find the area is:
(0,0), (3,2), (4,3), (0,0)
Area = , which is option .
Using Pick's Theorem, we can simply use the number of coordinate points to get the formula. In application, we get the formula , which equals , giving us our answer .
Using the Pythagorean Theorem, we know is . Using as a base of the triangle, the height is . The area would be , which equals , so the answer is .
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