Which sets of lengths can not be the sides of a triangle?

1 Expert Answer

Kemal G. answered • 03/15/17

Patient and Knowledgeable Math and Science Tutor with PhD

Hi Ericka,

You should make use of the Triangle Inequality Theorem, which states that

the sum of the lengths of two sides of a triangle must always be greater than the length of the third side

As you test the answers, you will notice that in C 7+11=18, which is also equal to the length of the other side. This is impossible so the answer is C.

To test each set, square all three numbers and see if the sum of the first two squares equals the sum of the third square.

Examples:
Does   #36^2 +   77^2#  equal   #85^2?#
Does #1296 + 5929# equal #7225?#
Since the squares of the two smaller sides do add up to the square of the longest side, then these are side lengths of a right triangle.

Does   #24^2 +   33^2#  equal   #42^2?#
Does #576 + 1089#  equal #1764?#
But in this case, the squares of the two smaller sides do not add up to the square of the longest side, so these are not the side lengths of a right triangle.
.................

Because all the answer choices are whole numbers, this is a question about "Pythagorean triples."

Here is what Wiki says about Pythagorean triples:
https://en.wikipedia.org/wiki/Pythagorean_triple

A Pythagorean triple
consists of three whole numbers#a, b,# and #c#
where #a^2 + b^2 = c^2##larr# the formula for the sides of a right triangle

#a# is always the smallest number and #c# is always the largest, making #c# the length of the hypotenuse of a right triangle

Triples are written #(a, b, c)#,
A famous example of a Pythagorean triple is #(3, 4, 5)#
This means that the two regular sides of the triangle are 3 units and 4 units long, and the hypotenuse is 5 units long.

If #(a, b, c)# is a right triangle, then all the multiples of that set are also right triangles. #(3, 4, 5)# is a right triangle, so #(6, 8, 10)# is too.
#3^2 + 4^2 = 5^2# and #6^2 + 8^2 = 10^2#

The side lengths of most right triangles are not Pythagorean triples. For example, #(1, 1, sqrt(2))# is a right triangle, but it's not a Pythagorean triple because its sides are not whole numbers.

The oldest known record of a Pythagorean triple comes from a Babylonian clay tablet from about 1800 BC.
..................

I would never waste my study time squaring and adding
twelve big numbers.

Instead, just look up the given #(a, b, c)# sets on this list.
https://en.wikipedia.org/wiki/Pythagorean_triple#Examples
The triples are listed in numerical order by the length of the hypotenuse.
........................

Standardized timed tests like the SAT, ACT, and GRE use Pythagorean triples all the time. If you know the sides by memory, you have a big advantage because you can avoid burning up your minutes on squaring.

Therefore, students should memorize the first three triples and be at least familiar with one or two more.

Can any 3 side lengths form a triangle?

For instance, can I create a triangle from sides of length...say 4, 8 and 3?

No!It's actually not possible!

As you can see in the picture below, it's not possible to create a triangle that has side lengths of 4, 8, and 3

It turns out that there are some rules about the side lengths of triangles. You can't just make up 3 random numbers and have a triangle! You could end up with 3 lines like those pictured above that cannot be connected to form a triangle.

Video On Theorem

The Formula

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.

Note: This rule must be satisfied for all 3 conditions of the sides.

In other words, as soon as you know that the sum of 2 sides is less than (or equal to) the measure of a third side, then you know that the sides do not make up a triangle.

Do I have to always check all 3 sets?

NOPE!

You only need to see if the two smaller sides are greater than the largest side!

Look at the example above, the problem was that 4 + 3 (sum of smaller sides) is not greater than 10 (larger side)

We start using this shortcut with practice problem 2 below.

Interactive Demonstrations of Theorem

The interactive demonstration below shows that the sum of the lengths of any 2 sides of a triangle must exceed the length of the third side. The demonstration also illustrates what happens when the sum of 1 pair of sides equals the length of the third side--you end up with a straight line! You can't make a triangle!

Otherwise, you cannot create a triangle from the 3 sides.

Mouseover To Start Demonstration

Practice Problems

Problem 1

Could a triangle have side lengths of

• Side 1: 4
• Side 2: 8
• Side 3: 2

No

Use the triangle inequality theorem and examine all 3 combinations of the sides. As soon as the sum of any 2 sides is less than the third side then the triangle's sides do not satisfy the theorem.

Problem 2

Could a triangle have side lengths of

• Side 1: 5
• Side 2: 6
• Side 3: 7

Yes

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

small + small > large

because 5 + 6 > 7

Problem 3

Could a triangle have side lengths of

• Side 1: 1.2
• Side 2: 3.1
• Side 3: 1.6

No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

small + small > large

because 1.2 + 1.6 $$\color{Red}{ \ngtr } $$ 3.1

Problem 4

Could a triangle have side lengths of

• Side 1: 6
• Side 2: 8
• Side 3: 15

No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

small + small > large

because 6 + 8 $$\color{Red}{ \ngtr } $$ 16

More like Problem 1-4...

Problem 4.1

Could a triangle have side lengths of

• Side 1: 5
• Side 2: 5
• Side 3: 10

No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

small + small > large

because 5 + 5 $$\color{Red}{ \ngtr } $$ 10

Problem 4.2

Could a triangle have side lengths of

• Side 1: 7
• Side 2: 9
• Side 3: 15

Yes

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

small + small > large

because 7 + 9 > 15

Practice Problems Harder

Problem 5

Two sides of a triangle have lengths 8 and 4. Find all possible lengths of the third side.

You can use a simple formula shown below to solve these types of problems:

difference $$< x <$$ sum
$$8 -4 < x < 8+4 $$

Answer: $$4 < x < 12$$

There's an infinite number of possible triangles, but we know that the side must be larger than 4 and smaller than 12 .

One Possible Solution

---

Here's an example of a triangle whose unknown side is just a little larger than 4:

Another Possible Solution

---

Here's an example of a triangle whose unknown side is just a little smaller than 12:

Problem 6

Two sides of a triangle have lengths 2 and 7. Find all possible lengths of the third side.

difference $$< x <$$ sum
$$7 -2 < x < 7+2$$

Answer: $$5 < x < 9$$

Problem 7

Two sides of a triangle have lengths 12 and 5. Find all possible lengths of the third side.

difference $$< x <$$ sum
$$12 -5 < x < 12 + 5$$

Answer: $$7 < x < 17$$

Related Links:

• Worksheet on remote, exteior and inteior angles of a Triangle
• Triangle Formulas
• Triangles
• Triangle Types
• Interactive Triangle
• Remote Interior Angles
• Area of a Triangle
• Triangle Inequality Theorem
• Free Triangle Worksheets

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