Question

If you can please do both problems it would be great (25)

Answer 1

Problem 25

Let a,b,c be the three sides of a triangle. They are positive real numbers.

A triangle is possible if and only if all 3 of these conditions are true:

• a+b > c
• a+c > b
• b+c > a

Phrased another way: pick any two sides and add them up. Their sum must be larger than the third side.

In this case we have a = 8, b = 9, c = 11

• a+b = 8+9 = 17 is larger than c = 11. So a+b > c is true
• a+c = 8+11 = 19 is larger than b = 9. This makes a+c > b true
• b+c = 9+11 = 20 is larger than a = 8. So b+c > a is true

We found all three inequalities to be true. Therefore, a triangle is possible due to the triangle inequality theorem.

Answer: A triangle is possible.

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Problem 26

Now we have

• a = 7.4
• b = 8.1
• c = 9.8

Let's use the same idea discussed in problem 25.

• a+b = 7.4+8.1 = 15.5 is larger than c = 9.8; a+b > c is true
• a+c = 7.4+9.8 = 17.2 is larger than b = 8.1; a+c > b is true
• b+c = 8.1+9.8 = 17.9 is larger than a = 7.4; b+c > a is true

Answer: A triangle is possible.

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Bonus Section:

Let's say we had these side lengths

• a = 1
• b = 5
• c = 10

a+b = 1+5 = 6 is NOT larger than c = 10, which means a triangle isn't possible for this example scenario.

I recommend making slips of paper 1 inch, 5 inches and 10 inches long. The 1 inch and 5 inch slips of paper come up short compared to the 10 inch side, which is why a full triangle is not possible here.

Answer 2

GIVEN

A triangle with sides 8 ft, 9 ft, and 11 ft.

SOLUTION

The triangle inequality theorem states that the sum of any two sides of a triangle is greater than or equal to the third side:

`a+b\ge c`

For the triangle with sides 8 ft, 9 ft, and 11 ft, the triangle inequality can be applied as follows:

`\begin{gathered} 8+9>11 \\ 8+11>9 \\ 9+11>8 \end{gathered}`

Therefore, the sides can form a triangle.