SOLUTION: is it possible for a triangle to have sides of 6 inches 12 inches and 17 inches?

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Question 1205301: is it possible for a triangle to have sides of 6 inches 12 inches and 17 inches?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52810) About Me  (Show Source):
You can put this solution on YOUR website!
.

The necessary and sufficient condition for a triangle with side lengths "a", "b" and "c" to exist

is fulfillment of all three "triangle inequalities"

    a + b > c,
  
    a + c > b,

    b + c > a.


In your case, all three triangle inequalities are satisfied - hence, such triangle does exist.


It can be constructed using a compass and a straightedge (a standard procedure).

Solved, with explanations.


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On triangle inequalities,  see the lesson
    - Points and Straight Lines basics
in this site,  Theorem 2.

You will be surprised to learn that triangle inequalities are very first corollaries from the basic axioms of Geometry.


On basic procedures of constructing triangles using a compass and a straightedge see the lesson
    - How to construct a triangle using a compass and a ruler
in this site.

Learn the subject from there.



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Use the Triangle Inequality Theorem to establish these 3 inequalities:
a+b > c
a+c > b
b+c > a
If all 3 inequalities are true, then a triangle is possible.

In this case we have: a = 6, b = 12, c = 17.
a+b > c
6+12 > 17
18 > 17
true		a+c > b
6+17 > 12
23 > 12
true		b+c > a
12+17 > 6
29 > 6
true

All 3 inequalities are true, so a triangle is possible when the sides are 6 inches, 12 inches, and 17 inches.
Adding any two sides produces a sum larger than the third side.

GeoGebra can be used to confirm the answer.
Alternatively you can use slips of paper, or pieces of string, to try it out yourself.

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Another example:
Is a triangle possible when a = 2, b = 3, c = 10?

a+b > c
2+3 > 10
5 > 10
false		a+c > b
2+10 > 3
12 > 3
true		b+c > a
3+10 > 2
13 > 2
true

The 1st inequality being false means a triangle is not possible when the sides are 2, 3, and 10.
We have 2 inequalities that are true, but we need ALL 3 of them to be true.
Technically you do not have to check the other inequalities once you reach a false inequality. But it could be good practice.