AMC 8 2019 Problem 4
AMC8 2019 年第 4 题
四边形与多边形★★☆☆☆
Quadrilateral ABCD is a rhombus with perimeter 52 meters. The length of diagonal is 24 meters. What is the area in square meters of rhombus ABCD ?
四边形 ABCD 是个周长为 52 米的菱形。对角线是多少平方米？
- A.60
- B.90
- C.105
- D.120
- E.144
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
A rhombus has sides of equal length. Because the perimeter of the rhombus is 52, each side is . In a rhombus, diagonals are perpendicular and bisect each other, which means = 12 = . Consider one of the = 13, and = 12. Using the Pythagorean theorem, we find that = 5. You know the Pythagorean triple, (5, 12, 13). Thus the values of the two diagonals are = 24 and = 10. The area of a rhombus is
Right off the bat, we can see that the perimeter of the figure is 52. Dividing this by four, we can get that each side is equal to 13. By drawing a line perpendicular to the one given, we can split the figure into four right triangles. 12 () is equal to the height of one small right triangle, and 13 is the slanted side. Using the Pythagorean theorem we can find that 169 (13 squared) - 144 (12 squared) = 25 (five squared). With this, we can determine that each small right triangle equals 30. Multiplying that by four we can get 120.
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