Geometry Worksheets

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Triangle Inequality Theorem

By Shrinivasa M

Learn about the Triangle Inequality Theorem and its use in geometry to solve related problems. Find the definition, example problems, and practice problems at Thinkster Math.

Why is this concept useful?

This theorem is mainly useful for identifying whether any 3 given lengths can form a triangle or not. It is used as a postulate for proving many other theorems in geometry, and also in solving geometric problems and solutions.

Where does this concept fit into the curriculum?

High School Geometry

What is the reflexive property of congruence?

The triangle inequality theorem states that for any given triangle, the sum of any two sides is always greater than the third side. It describes the relationship between the sides of the triangle and is applicable for any type of triangle be it equilateral, isosceles or scalene.

Consider a ∆ABC as shown where a, b, and c are the length of the sides.


According to the theorem ABC is a triangle only if,

a+b>c,

b+c>a,

c+a>b.

How can we use the triangle inequality theorem?

It is a very simple theorem to use. Whenever 3 lengths are given, we need to just verify if the sum of any 2 is always greater than the third side.

Suppose, we need to check 11,13 and 15 can form a triangle

11+13 >15

13+15>11

15+11>13

So, there exists a scalene triangle with the sides 11,13 and 15 units.

If side lengths are given to be 4, 4, and 8 units, these form 3 collinear points since,

4+4 = 8 (the triangle inequality theorem fails for one set as the sum should be greater than third)

So it's not possible to have a triangle with these dimensions. However, 5,5 and 8 are possible.

Note that in a right triangle with sides 3, 4, and 5 or any equilateral triangle too, this constraint will be met.



Sample Math Problems

Question 1: Can line segments of length 3 cm, 7 cm and 6 cm be used to form a triangle? Check if 4.1 cm, 6.9 cm and 2.81 cm measures, forms a triangle?

Answer:

3+7 > 6

7 +6 >3

6+3 >7

Yes they can form a triangle.

For the second instance,

4.1 + 6.9 = 11 >2.81

6.9 + 2.81 = 9.71 > 4.1

4.1 + 2.81 = 6.91 > 6.9

Yes they form a triangle.


Question 2: If 2 sides of a triangle are 5 and 8 units, find all possible lengths of the third side.

Answer:

Let the third side be x.

x< (5 +8) since 5+8 > x for triangle to exist.

x <13.

But 5+x>8 which means x>8-5 which is x>3

Also, 8+x>5 which means x>5-8 which is x>-3

x>3 and x>-3 implies that x>3

So, x can take any value from 3 to 13 units.


Question 3: Do the points (1,3) (3,5) and (5,7) form a triangle?

Answer:

We need to first find the length of the sides using the distance formula and then apply the inequality theorem.

AB = √(1-3)2 +(3-5)2 = √4+4 = 2.83

BC = √(3-5)2 +(5-7)2 = √4+4 = 2.83

AC = √(1-5)2 + (3-7)2 = √16+16 = 5.66

AB+BC = AC which does not satisfy the inequality theorem. All points are collinear.

Collinear points don't form a triangle.


Question 4: Is it possible to have a hypotenuse of an isosceles triangle double the length of the other 2 sides?

Answer:

No, because shorter sides of the triangles are equal in isosceles and if each is x,

Then x +x =2x and double the length of shorter sides is also 2x

But sum of any two sides > third side(hypotenuse).

So it is not possible to have such a triangle.


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Practice Math Problems

Question 1:

Can we have a triangle formed with all sides equal, where each side is of any real value? [Hint: x + x >x] Does this mean this is applicable for isosceles triangles too?

Question 2:

Check if a triangle can be formed whose sides are 7cm, 4 cm, and 2 cm?

Question 3:

What are the possible values of the third side of a triangle whose 2 sides are 5 cm each.

Question 4:

A person who was initially at point A(3,5) moved to point B(2,7) and then to point C(4, 9). Do A, B, and C form the vertices of a triangle?

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