The Triangle Inequality Theorem explains the relationship between the sides of a triangle and helps us determine whether a triangle can be formed using given side lengths.
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. This condition is necessary for the formation of a triangle.
For example, consider three lengths: 3 cm, 4 cm, and 5 cm.
Here, 3 + 4 = 7 > 5, 4 + 5 = 9 > 3, and 5 + 3 = 8 > 4.
Since all three conditions are satisfied, a triangle can be formed with these side lengths.
Triangle Inequality Theorem Proof
To prove the theorem, assume there is a triangle ABC in which side AB is produced to D and CD is joined.
Notice that the side BA of Δ ABC has been produced to a point D such that AD = AC. Now, since ∠BCD > ∠BDC.
By the properties mentioned above, we can conclude that BD > BC.
We know that, BD = BA + AD
So, BA + AD > BC
= BA + AC > BC
So, this proved sum of two sides triangle is always greater than the other side.
Let's see an example based on Triangle Inequality Theorem to understand its concept more clearly.
Example: D is a point on side BC of triangle ABC such that AD = DC. Show that AB > BD.
In triangle DAC,
AD = AC,
∠ADC = ∠ACD (Angles opposite to equal sides)
∠ ADC is an exterior angle for ΔABD.
∠ ADC > ∠ ABD
⇒ ∠ ACB > ∠ ABC
⇒ AB > AC (Side opposite to larger angle in Δ ABC)
⇒ AB > AD (AD = AC)
Triangle Inequality Theorem - Applications
There are many applications in the geometry of the Triangle Inequality Theorem, some of those applications are as follows:
- To Identify the Triangles
- To Find the Range of Possible Values of the Sides of Triangles
Identifying Triangles
To Identify the possibility of the construction of any given triangle with three sides, we can use the Triangle Inequality Theorem. If the given three sides satisfy the theorem, then the construction of this triangle is possible.
For example, consider the sides of the triangle as 4 units, 5 units, and 7 units.
4 + 5 > 7, 5 + 7 > 4, and 7 + 4 > 5. Thus, the triangle with sides 4 unit, 5 units, and 7 units is possible to construct.
Now take another example for not possible construction, consider the sides of the triangle to be 3 units, 4 units, and 9 units.
As 3 + 4
\ngtr 9, therefore triangle with sides 3 units, 4 units and 9 units is not possible.
Finding Range of Possible Values of Sides of Triangle
To find the range of possible values of the third side of a triangle when two sides are given as a units and b units, follow these steps:
Step 1: Assume the third side is x units.
Step 2: Using the Triangle Inequality Theorem:
a + b > x,
a + x > b,
b + x > aStep 3: From these inequalities, we get:
x < a + b
x > b − a
x > a − bCombining the last two conditions:
x > |a − b|So, the range of the third side is:
|a − b| < x < a + b
Example: Find the range for the third side of the triangle if the first two sides are 4 units and 7 units.
Let's assume the third side be x units.
Using Triangle Inequality Theorem, we get
4 + 7 = 11 > x, 4 + x > 7, and 7 + x > 4
Simplifying the above inequalities, we get
11 > x and x > 3.
Thus, possible range for the third sides is 3 < x < 11.
Sample Problems
Problem 1: Determine whether the given set of side lengths can form a triangle according to the triangle inequality theorem.
a) 3, 4, 9
b) 5, 7, 12
c) 6, 10, 25
For any triangle with sides a, b and c, using the triangle inequality theorem we get.
- a + b > c,
- b + c > a,
- c + a > b
We can use this to determine whether a triangle can be formed or not.
a) 3, 4, 9
As 3 + 4 < 9,
Thus, this triangle can't be formed.
b) 5, 7, 12
As 5 + 7 = 12,
Thus, this triangle also can't be formed.
c) 6, 10, 25
As 6 + 10 < 25,
Thus, this triangle also can't be formed.
All the given triangles do not satisfy the triangle inequality theorem.
Problem 2: If the two sides of a triangle are 3 and 5. Find all the possible lengths of the third side.
As we know, that for any triangle with sides a, b and c,
a - b < c < a + b
Let he unknown side be x,
Thus, 5 - 3 < x < 5 + 3
⇒ 2 < x < 8
Thus, third side can have any value between 2 and 8.
Practice Problem
Problem 1: Given a triangle with sides of lengths 5 cm, 8 cm, and 12 cm, determine whether the triangle satisfies the triangle inequality theorem.
Problem 2: Suppose you have a triangle with sides of lengths 7 inches, 10 inches, and 28 inches. Can this triangle exist?
Problem 3: Three sticks have lengths 9 cm, 15 cm, and 20 cm. Can we form a triangle using these sticks?