If two sides of a triangle have lengths 2 and 5 which of the following

Sum of two sides should be greater than the third side.
Difference between two sides should be lesser tha the third side.

So, the third side lies in the range

3 < c < 7

So perimeter lies in the range

10 < p < 14

Answer is A.

Here is my approach => the key here is to realise that x can have any value in the range => (3,7)
So none of the options are a MATCH => A

We are given that two sides of a triangle have lengths 2 and 5, and we need to determine the possible perimeter of the triangle. Here we are being tested on the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, the sum of the lengths of the two shortest sides must be greater than the length of the longest side. Let’s evaluate the Roman numeral answer choices.

I. perimeter = 9

The two known sides sum to 7. If the perimeter is 9, then the 3rd side is 9 – 7 = 2, giving us sides of 2, 2, and 5. Since 2 + 2 = 4 is less than 5, 9 cannot be the perimeter of the triangle.

II. perimeter = 15

If the perimeter is 15, then the 3rd side is 15 – 7 = 8, giving us sides of 2, 5, and 8. Since 2 + 5 = 7 is less than 8, 15 cannot be the perimeter of the triangle.

III. perimeter = 19

If the perimeter is 19, then the 3rd side is 19 – 7 = 12 giving us sides of 2, 5, and 12. Since 2 + 5 = 7 is less than 12, 19 cannot be the perimeter of the triangle.

Answer: A

Bunuel, what about the case when triangle is isosceles. Why Cann't we consider 2,2 and 5 as sides or 2,5,5 sides of triangle.

Thanks,
Amm

1. Sides of 2, 2, 5 won't create a triangle: 2 + 2 < 5.

2. The perimeter of {5, 5, 2} triangle is 12, which is in the range we got: 10 < {perimeter} < 14.

the third side of ∆ would be 3<third side<7
sum of other two sides is 7
so 7+third side ; 7+4; 7+5;7+6
11,12,13
which is not given options
hence option A ; none

Once you start reading the question, the first thing to strike you should be the theorem: Sum of any two sides of a triangle is greater than 3rd side.

After this, the question becomes simple enough.If the two sides are 2 and 5, the 3rd side cannot be equal to or greater than 7. Hope that helps!

two sides are 2 and 5....hence the 3rd side will be less than 7 (5+2) and more than 3 (5-2) .


Hence the third side z .....3<z<7

hence perimeter should be between.....10<P<14

Option A

A triangle has 3 sides,let’s say x,y and z
There is a rule that states x+y > z
x+z > y
z+y > x
So the sum of two sides of a triangle is always bigger than the other.
So from this,to solve the question we have the lengths 2 and 5 so we know that we need an option smaller than 2+5 = 7
There is no option smaller than 7
Answer = A

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