Unlock the Mystery: Which of These Possibilities Forms a Triangle?
In the realm of geometry, triangles have captivated mathematicians for centuries. But what if you could create a triangle out of thin air? Imagine being able to determine which combinations of numbers or segments will form a triangle and which will leave you with a disconnected mess. This blog post will unveil the secrets behind this geometric puzzle, empowering you to form triangles like a pro!
The concept of forming triangles may seem straightforward, but it often leads to headscratching moments when dealing with complex possibilities. Misconceptions and assumptions can cloud our judgment, leaving us unsure whether our creations will result in a valid triangle.
To master the art of triangle formation, we must first understand the fundamental rules that govern them. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, the shorter sides must add up to more than the longest side. Equivalently, the difference between the lengths of any two sides must be less than the length of the third side. Let’s dive into specific examples to illustrate these rules and shed light on the possibilities that will form a triangle.
By applying the Triangle Inequality Theorem, we can determine which possibilities will lead to a valid triangle:
 3, 4, and 5 will form a triangle because 3 + 4 > 5 and 3 + 5 > 4 and 4 + 5 > 3.
 2, 5, and 7 will form a triangle because 2 + 5 > 7 and 2 + 7 > 5 and 5 + 7 > 2.
 1, 6, and 8 will not form a triangle because 1 + 6 = 7 < 8.
Exploring Triangle Formation: Prerequisites and Possibilities
In geometry, a triangle is a polygon with three sides and three angles. To determine whether a set of three given possibilities will form a triangle, we must consider specific conditions that ensure triangle formation.
Basic Triangle Formation Criteria
For three possibilities (a
, b
, and c
) to form a triangle, they must satisfy the following criteria:

Triangle Inequality Theorem:
a + b > c
,b + c > a
, andc + a > b

SideSum Property: The sum of any two sides of a triangle must be greater than the length of the third side, i.e., for sides
a
,b
, andc
,a + b > c
,b + c > a
, andc + a > b
.
Possibilities that Form Triangles
Based on these criteria, the following possibilities will always form triangles:

Three Positive Numbers: If
a
,b
, andc
are all positive numbers, they satisfy the Triangle Inequality Theorem. 
Two Positive Numbers and a Zero: If two of the possibilities (
a
andb
, for example) are positive numbers and the third (c
) is zero, they also satisfy the Triangle Inequality Theorem. 
One Positive Number and Two Zeros: If only one possibility (
a
, for example) is a positive number and the other two (b
andc
) are zeros, the SideSum Property holds true.
Possibilities that Do Not Form Triangles
The following possibilities will not form triangles:

Three Zeros: The SideSum Property cannot be satisfied with three zeros.

Two Zeros and a Negative Number: The Triangle Inequality Theorem cannot be satisfied with two zeros and a negative number.

One Positive Number and Two Negative Numbers: The Triangle Inequality Theorem is violated when one positive number is paired with two negative numbers.
Special Cases
In certain scenarios, the possibility of triangle formation may depend on additional factors:

Two Equal Zeros and a NonZero: If two possibilities (
b
andc
) are equal to zero, and the third (a
) is nonzero, triangle formation depends on the value ofa
. Ifa
is greater than or equal to the sum of the two zeros, a triangle can be formed. 
Two Equal NonZeros and a Zero: If two possibilities (
a
andb
) are equal nonzeros, and the third (c
) is zero, a triangle is always formed.
Conclusion
Understanding the principles of triangle formation is crucial for geometry and other mathematical applications. By adhering to the criteria outlined, we can confidently determine which possibilities will and will not form triangles.
Frequently Asked Questions (FAQs)
 Can a triangle be formed with two sides equal to 5 cm and the third side equal to 12 cm?
 No, the Triangle Inequality Theorem is violated because 5 cm + 5 cm = 10 cm, which is not greater than 12 cm.
 Can a triangle be formed with sides 0, 0, and 5 cm?
 Yes, the SideSum Property holds true, and the Triangle Inequality Theorem is not violated.
 Can a triangle be formed with sides 3 cm, 4 cm, and 7 cm?
 No, the Triangle Inequality Theorem is violated because 3 cm + 4 cm = 1 cm, which is not greater than 7 cm.
 Can a triangle be formed with sides 2 cm, 2 cm, and 1 cm?
 Yes, the SideSum Property and Triangle Inequality Theorem are both satisfied.
 Can a triangle be formed with sides 0, 2 cm, and 2 cm?
 Yes, this is a special case where two possibilities are equal to zero and the third possibility is nonzero. Triangle formation depends on the value of the nonzero side, which in this case is 2 cm, satisfying the SideSum Property.
.
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