Which of the following can be the ratio of sides of a triangle?

Question

I.  3:\sqrt{4}:5 
II.  \sqrt{3}:4:5 
III.  3:4:\sqrt{5}

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

KarishmaB · Accepted Answer

The sum of any two sides of a triangle must be greater than the third side.

I.  3:\sqrt{4}:5 
The ratio of sides are 3, 2 and 5
But 3 + 2 = 5 which is not acceptable. The sum of any two sides must be greater than the third side. 
Hence these cannot be the ratio of the sides of the triangle.

II.  \sqrt{3}:4:5

The ratio of sides are 1.7, 4 and 5.
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

III.  3:4:\sqrt{5}

The ratio of sides are 3, 4 and 2.2
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

Answer (D)

Anigr16 · Answer

For any triangle with sides a,b,c, the following rules for sides MUST be followed:

|a-b| << c << a+b (this has to be true for ALL the sides!)

II and III: both satisfy the conditions
Hence, Ans: D

KGump · Answer

VeritasPrepKarishma ,  Bunuel ,

Is it sufficient to just check for condition - The sum of any two is greater than the third.

or do we also have to check that the difference of any two sides should be less than the third side

Thanks

KarishmaB · Answer

These are just different ways of saying the same thing.

a + b > c
or
a > c - b

Say sides are 2, 3 and 6.
Sum of 2 and 3 is not greater than 6.
Also difference of 6 and 3 is not less than 2.

KGump · Answer

Oh.. i see now.. always had this stupid doubt..
Thankyou for clearing this

dcummins · Answer

Official Explanation :
 
Solution: The sum of any two sides of a triangle must be greater than the third side.

4–√=2
. If the ratio of the sides is 3:2:5, the triangle cannot be formed because 3+2 is equal to 5, not greater.
3–√≈1.73
 (however, all we need to know is that 3–√
 is larger than 1 and smaller than 2). If the ratio of the sides is 1.73:4:5, the sum of any two sides will be greater than the third side. So the triangle can be formed.
5–√≈2.2
 (again, we just need to know it is larger than 2 but smaller than 3). If the ratio of the sides is 3:4:2.2, the sum of any two sides will be greater than the third side. So the triangle can be formed.

Fdambro294 · Answer

When given 3 Sides of the Triangle, one can apply the Rule that all you have to check is the following:

Shorter Side + Shorter Side > LONGEST Side

If the 3 Sides fulfill this Inequality, then the Triangle Inequality Theorem will be Satisfied and the Triangle is Valid.

Square Root of 4 = 2
Square Root of 3 = approx. 1.73
Square Root of 5 = approx. 2.24

I. 3 and 2 are the Shorter Sides. 5 is the Longest Side.

3 + 2 = 5. NOT a Valid Triangle.

II. 1.73 and 4 are the Shorter Sides. 5 is Longest Side

1.73 + 4 > 5. Valid Triangle.

III. 2.24 and 3 are the Shorter Sides. 4 is the Longest Side.

2.24 + 3 > 4. Valid Triangle.

Answer D: II and III only

EDIT: its the same idea as the Triangle Inequality Theorem discussed above, but approaching the question this way might save some seconds.

Basshead · Answer

The sum of two sides of a triangle must be greater than the third side.

1) 3: 2: 5

3 + 2 = 5; NOT POSSIBLE.

2) 1.7: 4: 5; POSSIBLE

3) 3: 4: 2.2; POSSIBLE

Therefore only II and III are possible. Answer is D.

srishR · Answer

For this question why can't we square all sides and try and compute?? I tried that and i wasn't able to get an answer
 VeritasKarishma   Bunuel

KarishmaB · Answer

When you square a ratio, it changes. It does not stay the same.

e.g. 
1:2
When you square it, it becomes 1 : 4.
This is a different ratio.

For example,  \sqrt{3}:4:5  will work as sides of a triangle but 3:16:25 will not.

bumpbot · Answer

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Which of the following can be the ratio of sides of a triangle?

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Which of the following can be the ratio of sides of a triangle?

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