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  3. If the lengths of two sides of a triangle are 5 and 9, respectively, which of th

If the lengths of two sides of a triangle are 5 and 9, respectively, which of th

游客2025-02-24  54
问题 If the lengths of two sides of a triangle are 5 and 9, respectively, which of the following could be the length of the third side of the triangle?
Indicate all such lengths.
选项 A、3
B、5
C、8
D、15
答案 B，C
解析 A good way to approach this problem is to think about how much the length of the third side of a triangle with two fixed side lengths can vary. If you think about it a bit, you will see that the smaller the interior angle between the two sides of the triangle is, the smaller the length of the third side is; and the larger the interior angle between the two sides of the triangle is, the larger the length of the third side is. This suggests drawing two triangles, one in which the angle between the two sides is close to 0 degrees and one in which the angle between the two sides is close to 180 degrees, like the triangles below.

    In the triangle in which the angle between the sides of length 5 and 9 is small, you can see that the length of the third side is a bit greater than 9 - 5, or 4. If it were equal to 4, the triangle would degenerate into a line segment.
    In the triangle in which the angle between the sides of length 5 and 9 is large, you can see that the length of the third side is a bit less than 9 + 5, or 14. If it were equal to 14, the triangle would degenerate into a line segment.
    Therefore, the length of the third side of the triangle must be greater than 4 and less than 14. Furthermore, it is intuitive that any length between these two numbers can be achieved by some triangle. The correct answer consists of Choices B and C.
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