AMC8 2010

AMC8 2010 · Q13

AMC8 2010 · Q13. It mainly tests Linear equations, Area & perimeter.

The lengths of the sides of a triangle measured in inches are three consecutive integers. The length of the shortest side is 30% of the perimeter. What is the length of the longest side?

一个三角形的边长（英寸）是三个连续整数。最短边长是周长的30%。最长边有多长？

(A) 7 7

(B) 8 8

(D) 10 10

(E) 11 11

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let $n$, $n+1$, and $n+2$ be the lengths of the sides of the triangle. Then the perimeter of the triangle is $n + (n+1) + (n+2) = 3n+3$. Using the fact that the length of the smallest side is $30\%$ of the perimeter, it follows that: $n = 0.3(3n+3) \Rightarrow n = 0.9n+0.9 \Rightarrow 0.1n = 0.9 \Rightarrow n=9$. The longest side is then $n+2 = 11$. Thus, answer choice $\boxed{\textbf{(E)}\ 11}$ is correct.

设边长为$n$、$n+1$、$n+2$。则周长为$n + (n+1) + (n+2) = 3n+3$。最短边长为周长的30%，有： $n = 0.3(3n+3) \Rightarrow n = 0.9n+0.9 \Rightarrow 0.1n = 0.9 \Rightarrow n=9$。最长边为$n+2 = 11$。因此，答案$\boxed{\textbf{(E)}\ 11}$正确。

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