The Triangle inequality theorem can be used to find the range of the third side of the triangle. The correct option is B, D, C, and A.
What is the triangle inequality theorem?
The Triangle inequality theorem of a triangle says that the sum of any of the two sides of a triangle is always greater than the third side.
Suppose a, b and c are the three sides of a triangle. Thus according to this theorem,
(a+b) > c
(b+c) > a
(c+a) > b
To know the range in which the third side of the triangle can be we will use the law of triangle inequality.
1.)
If the lengths of two sides of a triangle are 3 and 5, the length of the third side can be,
3 + 5 > x
8 > x
x + 3 > 5
x >2
Therefore, the value of x should be greater than 2 and less than 8.
Hence, the length of the third side cannot be 9 units. The correct option is B.
2.)
If the lengths of two sides of a triangle are 2 and 6, the length of the third side can be,
2 + 6 > x
8 > x
x + 2 > 6
x > 4
Therefore, the value of x should be greater than 4 and less than 8.
Hence, the length of the third side cannot be 4 units. The correct option is D.
3.)
If the lengths of two sides of a triangle are 5 and 9, the length of the third side can be,
5 + 9 > x
14 > x
x + 5 > 9
x > 4
Therefore, the value of x should be greater than 4 and less than 14.
Hence, the length of the third side can be 7 units. The correct option is C.
4.)
If the lengths of two sides of a triangle are 8 and 13, the length of the third side can be,
8 + 13 > x
21 > x
x + 8 > 13
x >5
Therefore, the value of x should be greater than 5 and less than 21.
Hence, the length of the third side can be 6 units. The correct option is A.
Learn more about the Triangle Inequality Theorem here:
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