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Re: If x is a positive integer, which of the following COULD
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26 May 2018, 21:09
There are some triangle rules that apply for all triangles. One of that is the length of a side of a triangle is always less than the sum of other two sides of the triangle.
Let us test option i
x, 2x +2, x+2
If we add the first and the third side x + x +2 = 2x+2 which is equal to the second side so this is false
Let us test option ii 2x, 3x, 2x - 7
Since the third side of this triangle includes a constant we can get some scenarios where the triangle can satisfy triangle properties and in some instance it does not satisfy the triangle property. However, this is a could be question so a single possibility is good enough to be true.
For eg. If the said integer is 1 we would have the three sides as, 2,3, -5 since length cannot be -ve this is not a possibility However let us take x = 10 then, 20, 30, 13 This satisfys the triangle property
Let us test option iii \(\frac{x}{2}, \frac{x}{6}, \frac{x}{4}\) The two smallest sides are x/6 and x/2. Option iii can only be a triangle if x/2 < x/6 + x/4 \(\frac{x}{6} + \frac{x}{4} = \frac{2x + 3x}{12} = \frac{5x}{12}\) Since \(\frac{x}{2}\) > \(\frac{5x}{12}\) option iii also is not correct.
A) i only B) ii only C) iii only D) i and ii only E) ii and iii only
There's a triangle property that says: (length of LONGEST side) < (SUM of the other two lengths) So, for example, 2, 3, 6 CANNOT be the lengths of sides in a triangle, because 6 > 2 + 3
i) x, 2x + 2, x + 2 Since x is a positive integer, we can see that 2x+2 will be the LONGEST side. Now compare this length to the SUM of the two other side lengths. Is it true that 2x + 2 < x + (x + 2)? Simplify: 2x + 2 < 2x + 2 This is NOT true. So, x, 2x + 2, x + 2 CANNOT represent the lengths of the 3 sides of a triangle Check the answer choices.... eliminate A and D
ii) 2x, 3x, 2x - 7 Since x is a positive integer, we can see that 3x will be the LONGEST side. Now compare this length to the SUM of the two other side lengths. Is it true that 3x < 2x + (2x - 7)? Simplify: 3x < 4x - 7 This COULD be true. For example, if x = 10, then we get: 3(10) < 4(10) - 7, which IS true. So, 2x, 3x, 2x - 7 COULD represent the lengths of the 3 sides of a triangle Check the answer choices.... eliminate C
iii) x/2, x/6, x/4 In this case, x/2 will be the LONGEST side. Now compare this length to the SUM of the two other side lengths. Is it true that x/2 < x/6 + x/4? Let's eliminate the fractions to make is easier for us to determine whether the inequality holds true. Multiply both sides of the inequality by 12 to get: 6x < 2x + 3x Simplify: 6x < 5x Since x is POSITIVE, we can see that this inequality is NOT true. So, x/2, x/6, x/4 CANNOT represent the lengths of the 3 sides of a triangle Check the answer choices.... eliminate E
Answer: B
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