The sum of the lengths of any two side of a triangle is greater than the length of the third side

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Geometry is the branch of mathematics that deals with the study of different types of shapes and figures and sizes. The branch of geometry deals with different angles, transformations, and similarities in the figures seen.

Triangle

A triangle is a closed two-dimensional shape associated with three angles, three sides, and three vertices. A triangle associated with three vertices says A, B, and C is represented as △ABC. It can also be termed as a three-sided polygon or trigon. Some of the common examples of triangles are signboards and sandwiches.

To prove: The sum of two sides of a triangle is greater than the third side, BA + AC > BC
Assume: Let us assume ABC to be a triangle.
Proof:
Extend the line segment BA to D,
Such that, AD = AC
⇒ ∠ ADC = ∠ ACD
Observing by the diagram, we obtain,
∠ DCB > ∠ ACD
⇒ ∠ DCB > ∠ ADC
⇒ BD > AB (Since the sides opposite to the larger angle is larger and the sides opposite to smaller angle is smaller)

⇒ BA + AD > BC
⇒ BA + AC > BC.
Hence proved.
Note: Similarly it can be also proved that, BA + BC > AC or AC + BC > BA
Hence, The sum of two sides of a triangle is greater than the third side.

Question 1. Prove that the above property holds for the lowest positive integral value.

Solution:

Let us assume ABC to be a triangle.
Each of the sides is 1 unit.
Now,
It is an equilateral triangle where all the sides are 1 each.
Taking sum of two sides,
AB + BC ,
1 + 1 > BC
1+1 > 1
2 > 1

Question 2. Illustrate this property for a right-angled triangle

Solution:

Let us assume the sides of the right angles triangle to be 5,12 and 13.
Now,
Taking the smaller two sides, we obtain,
5 + 12 > 13
17 > 13
Hence, the property holds.

Question 3. Does this property hold for isosceles triangles?

Solution:

Let us assume a triangle with sides 2x, 2x, and x.
Now,
Taking the sum of equal two sides, we obtain,
2x + 2x = 4x
which is greater than the third side, equivalent to x.

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

In the figure, the following inequalities hold.

a + b > c

a + c > b

b + c > a

Example 1:

Check whether it is possible to have a triangle with the given side lengths.

7 , 9 , 13

Add any two sides and see if it is greater than the other side.

The sum of 7 and 9 is 16 and 16 is greater than 13 .

The sum of 9 and 13 is 21 and 21 is greater than 7 .

The sum of 7 and 13 is 20 and 20 is greater than 9 .

This set of side lengths satisfies the Triangle Inequality Theorem.

These lengths do form a triangle.

Example 2:

Check whether the given side lengths form a triangle.

4 , 8 , 15

Check whether the sides satisfy the Triangle Inequality Theorem.

Add any two sides and see if it is greater than the other side.

The sum of 4 and 8 is 12 and 12 is less than 15 .

This set of side lengths does not satisfy Triangle Inequality Theorem.

These lengths do not form a triangle.

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Answer:

Triangle Inequality Theorem

Step-by-step explanation:

The sum of the lengths of any two sides of a triangle must be greater than the third side. This Theorem is called the Triangle Inequality Theorem. If these inequalities are NOT true, you will not have a triangle! There are two other "inequalities" that pertain to triangles that are "common sense" concepts.