Question

一个三角形的三条边中有两条分别为 $20$ 和 $15$ 。下列哪个数不可能是这个三角形的周长？

Two of the three sides of a triangle are $20$ and $15.$ Which of the following numbers is not a possible perimeter of the triangle?

$52$

$57$

$62$

$67$

$72$

第三边严格介于 $20-15$ 和 $20+15$ 之间

The third side lies strictly between $20-15$ and $20+15$

把 $20+15$ 加到这个范围上，就能得到周长的范围

Add $20+15$ to that range to bound the perimeter

根据三角形不等式，第三边 $s$ 满足 $20-15 \lt s \lt 20+15$ ，也就是 $5 \lt s \lt 35$ 。

周长为 $35 + s$ ，所以它严格介于 $40$ 和 $70$ 之间。选项中只有 $72$ 落在这个范围之外。

因此，正确答案是 E。

By the Triangle Inequality, the third side $s$ satisfies $20-15 \lt s \lt 20+15,$ that is $5 \lt s \lt 35.$

The perimeter is $35 + s,$ so it lies strictly between $40$ and $70.$ Among the choices, only $72$ falls outside this range.

Thus, the correct answer is E.