Question

❤❤❤❤❤❤❤❤❤❤ state and prove triangle inequalities theorem ❗️❗️❗️

Answer 1

Triangle Inequality Theorem

The triangle inequality theorem states that the length of any of the sides of a triangle must be shorter than the lengths of the other two sides added together. This tells us that in order for three line segments to create a triangle, it must be true that none of the lengths of each of those line segments is longer than the lengths of the other two line segments combined.

For example, let's look at our initial example. We were able to create a triangle with line segments having lengths 3, 4, and 5 units. This is because those line segments satisfy the triangle inequality theorem.

3 + 4 = 7 and 5 < 74 + 5 = 9 and 3 < 93 + 5 = 8 and 4 < 8

We see that none of the line segments are longer than the other two line segments combined.

However, if we consider the line segments with lengths 3, 4 and 9, we see that the line segment with length 9 units is longer than the other two line segments combined.

3 + 4 = 7 and 9 > 7

This explains why we couldn't create a triangle with these three line segments. They don't satisfy the triangle inequality theorem.

Answer 2

Draw a triangle, △ ABC and line perpendicular to AC passing through vertex B.

Proof of Triangle Inequality Theorem
Prove that BA + BC > AC

From the diagram, AM is the shortest distance from vertex A to BM.

and CM is the shortest distance from vertex C to BM.

i.e. AM < BA and CM < BC

By adding these inequalities, we have

AM + CM < BA + BC

=> AC < BA + BC ( ∵ AM + CM = AC)

BA + BC > AC (Hence Proved)

Similarly, we can solve this theorem for another sides.