SOLUTION: The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a

Question 1200184: The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a triangle to be able to use it in his design.
Option 1: Side lengths: 4, 4, 8
Option 2: Side lengths: 6, 8, 10
Option 3: Side lengths: 6, 6, 13
Part A. Which of the options would create a triangle for his design?
Part B. The homeowner would like the porch to be in the shape of a right triangle. Will the carpenter be able to use any of the given options?
Part C. For any option that does not form a triangle, what side length could be changed to
form a triangle? Explain your answer.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52777) (Show Source): You can put this solution on YOUR website!
.

According to triangle inequalities, options (1) and (3) do not create a triangle.

Triangle (2) is a right-angled triangle: you can check it on your own,
using the Pythagorean theorem.

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

Part A

If a triangle has sides a,b,c then all of the following conditions must be true
a < b+c
b < a+c
c < a+b
Any two sides must add to be larger than the third side.
See the triangle inequality theorem.

In option 1, we have: a = 4, b = 4, c = 8
And a+b = 4+4 = 8 which does not exceed c = 8
In other words, c < a+b is not true.
A triangle isn't possible here.
Option 3 is a similar story.

Option 2 on the other hand works
b+c = 8+10 = 18 exceeds a = 6 so a < b+c is true.
a+c = 6+10 = 16 exceeds b = 8 so b < a+c is true.
a+b = 6+8 = 14 exceeds c = 10 so c < a+b is true.
A triangle is possible with sides 6, 8, and 10.

==================================================
Part B

Use the pythagorean theorem
Note that a = 6, b = 8, c = 10 satisfy this equation.
Therefore, option 2 represents a right triangle.

==================================================
Part C

Let's look at option 1
Let's say we knew the first two sides a = 4 and b = 4
For a triangle to be possible, the third side c is given by this range
b-a < c < b+a
where

See this my answer in this post for more info
https://www.algebra.com/algebra/homework/Triangles/Triangles.faq.question.1200226.html

The range for c in this case would be
b-a < c < b+a
4-4 < c < 4+4
0 < c < 8

This means we could pick something like c = 5
A triangle is possible with sides a = 4, b = 4, c = 5

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