SOLUTION: Benjamin is drawing a triangle and knows that two sides are 9 centimeters and 12 centimeters. What could the third side length be? Select all that apply 2 cm 4 cm 10 cm

Question 1200226: Benjamin is drawing a triangle
and knows that two sides are
9 centimeters and 12 centimeters.
What could the third side length be?
Select all that apply
2 cm
4 cm
10 cm
18 cm
21 cm
Found 3 solutions by ikleyn, greenestamps, math_tutor2020:
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.

Triangle inequalities say that the third side of the triangle is shorter than the sum of the two other sides third side is shorter than 9 + 12 = 21 cm. The same inequalities say that the third side of the triangle is longer than the difference of the two other sides third side is longer than 12 - 9 = 3 cm. It leaves possible options 4 cm, 10 cm and 18 cm and excludes other options for the length of the third side.

Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!

In any triangle, the sum of the lengths of the two shorter sides must be greater than the length of the longest side.

If the side with length 12cm is the longest, then the missing side must have length x where 9 plus x is greater than 12.

If the missing side is the longest side, then its length x must be such that 9 plus 12 is greater than x.

Now you answer the question.

Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answers:
4 cm
10 cm
18 cm

=============================================================================

Explanation:

We have a triangle with sides a,b, and c.

a = 9 and b = 12 are the two known sides.
c is the unknown side.

Due to the triangle inequality theorem, we have these three conditions that must all be true if we want a triangle to be possible
a < b+c
b < a+c
c < a+b
Rephrased verbally: "The sum of any two sides must exceed the third side".

Let's isolate c in the 1st two inequalities
a < b+c becomes a-b < c
b < a+c becomes b-a < c

If b ≥ a, then we'll focus on b-a < c so that the left hand side is nonnegative.

At the same time, the third inequality condition of the triangle inequality theorem states that c < a+b or c < b+a

We have these inequalities
b-a < c and c < b+a
they can be conveniently glued together to get
b-a < c < b+a
This gives a range of possible c values where b ≥ a.

---------------------------------------------

Summary of the previous section:

Given a triangle with sides {a,b,c} where a & b are known lengths, and b ≥ a, the third side c has the restrictions of: b-a < c < b+a

From here we plug in a = 9 and b = 12
b-a < c < b+a
12-9 < c < 12+9
3 < c < 21

The third side is between 3 cm and 21 cm, excluding each endpoint.
We cannot have c = 3. We also cannot have c = 21.
These endpoints will cause a straight line to form rather than a triangle.

The following answer choices are in the interval mentioned:
4 cm
10 cm
18 cm
which are the final answers.

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