Geometry Question

[dongkim2]

If the four sides of a pentagon is 2, 3, 4, 5, respectively, which of the following could be the length of the fifth side?

I. 5
II. 10
III. 15

A None
B I only
C III only
D I and II only
E I, II, and III all


OA D

Please explain. Thanks!

[rockeyb]

If the four sides of a pentagon is 2, 3, 4, 5, respectively, which of the following could be the length of the fifth side?

I. 5
II. 10
III. 15

A None
B I only
C III only
D I and II only
E I, II, and III all


OA D

Please explain. Thanks!

OK here it goes a bit tricky one .

A pentagon is a closed figure just like a triangle .

In order for a polygon to be a closed figure sum of all the sides must be greater than any one side of the polygon .

Ex : In a triangle with sides 2, 3, 5 .

Sum of any two side is always greater than the third sides .

But if one of the sides is 7 then this figure will not be a triangle OR the figure will not be a close figure.

Similarly in a pentagon with sides 2, 3, 4, 5 .

Sum of the sides = 2+3+4+5 = 14 .

So 14 is the max length of the 5 th side or its the limiting value for the figure to be a closed figure .

Hence only possible values are 10 and 5 .

But I think we need to consider the lower limit value for the 5 th side .

To calculate that we need to find the difference between the largest side in this case 5 and sum of the rest of the sides .

So (2+3+4)- 5 = 4 .

So the 5 th side must be greater than 4 and less than 14.

Option D is correct.