If 3 and 8 are the lengths of two sides of a triangular region, which

I don't quite understand what you mean by the red part above. Anyway:

If 3 and 8 are the lengths of two sides of a triangular region, which of the following can be the length of the third side?

I. 5
II. 8
III. 11

(A) ΙI only
(B) ΙII only
(C) I and ΙI only
(D) II and ΙII only
(E) I, ΙΙ, and ΙΙI

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

So, \((8-3)<x<(8+3)\) --> \(5<x<11\). Hence 5 and 11 cannot be the length of the third side, while 8 can be.

Answer: A.

For more check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.
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the rest side x must satisfy : 8-3<x< 8+3 <==> 5<x<11
B.

the sum of the two sides must be greater than the third side. 3+8 =11
the difference of the two sides mut be less than the third side. 8-3=5

therefore the third side is between 5 and 11. Only B fulfills this.

Can't we assume 5<x<11? what if it's an iscoceles triangle, in which case the third side could be equal to either of the two sides?

The only possibility for the third side of the triangle is 8
because the property of the triangle which is used over here is
the sum of any 2 sides must be larger than the third side.

Hence Option A(II only) is the answer choice

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.
Only possible value is 8 as 5<third side<11

A
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The third side of a triangle is always shorter than the sum of the other two sides:

x < 3 + 8
x < 11

AND the third side must be greater than the difference between the other two sides

x > 8 - 3
x > 5
5 < x

Together:

5 < x < 11

Third side x cannot be 5 or 11.

Only Option II satisfies the triangle inequality theorem.

Answer A
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Hi All,

We're told that 3 and 8 are the lengths of two sides of a triangular region. We're asked which of the following values (3, 8 and 11) could be the length of the third side. This question is based on the Triangle Inequality Theorem and requires just a little Arithmetic to solve.

To start, when you have two sides of a triangle, the SMALLEST the third side could be would be just a little bit MORE than the DIFFERENCE of the two sides you have. Here, that's 8 - 3 = 5. If the third side was EXACTLY 5, then you would NOT have a triangle - you'd have a line of 8 on top of another line of 8. By making that third side just a little bigger than 5, you would then have a triangle.

In that same way, the LARGEST the third side could be would be just a little bit LESS than the SUM of the two sides. Here, that's 3 + 8 = 11. If the third side was EXACTLY 11, then you would NOT have a triangle - you'd have a line of 11 on top of another line of 11. By making that third side just a little less than 11, you would then have a triangle.

Thus, that third side falls into the following range: 3 < third side < 11. Based on the three given options, only a side of 8 is possible.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
difference between sides A and B < third side < sum of sides A and B

So, for this question: 8 - 3 < third side < 8 + 3
Simplify: 5 < third side < 11

So, the third side must be LONGER than 5 and SHORTER than 11
Answer: A

Cheers,
Brent

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