SAT Geometry : Scalene Triangles

Question 
If two sides of a scalene triangle have lengths 3 and 5, the perimeter of the triangle could be which of the following?
I. 13
II. 14
III. 16

A) I only
B) II only
C) III only
D) II and III only
E) I and II only

Correct Answer : Choice B. II only.

Explanatory Answer

The triangle mentioned in the question is a scalene triangle.
i.e., the measure all 3 sides of the triangle are different.

Two of its sides are given as 3 and 5. So, the sum of these two sides = 8.

We need to find out which among the 3 answers could be the perimeter of the triangle with the above information.

Option I : Perimeter 13 units.

If the perimeter is 13 units, the measure of the 3rd side of the triangle = 13 - 8 = 5.
So, the sides of the triangle will be 3, 5 and 5.
The triangle with two equal sides is an isosceles triangle and not a scalene triangle.
So, 13 CANNOT be the perimeter of the triangle.

Option II : Perimeter 14 units.

If the perimeter is 14 units, the measure of the third side of the triangle = 14 - 8 = 6.
So, the sides of the triangle will be 3, 5 and 6.
The triangle is a scalene triangle.
So, 14 can be the perimeter of the triangle

Option III : Perimeter 16 units.

If the perimeter is 16 units, the measure of the third side = 16 - 8 = 8.
So, the sides of the triangle will be 3, 5 and 8.

One of the basic properties of a triangle is that sum of any two sides is greater than the third side.
In this case, sum of 3 and 5 i.e., 3 + 5 = 8 and NOT greater than 8.
So, 3, 5 and 8 will not form sides of a triangle.

Therefore, 16 CANNOT be the perimeter of the triangle.

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