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Chapter 5: Problem 36

Suppose one side of a triangle has length 5 and another side of the triangle has length 8 . Let \(c\) denote the length of the third side of the triangle. Show that \(3

Short Answer

Expert verified
Applying the triangle inequality theorem to the given triangle sides, we get the inequalities \(13>c\) and \(c>3\). Therefore, the length of the third side (c) must be within the range \(3<c<13\).

Step by step solution

01

Apply the triangle inequality theorem to Side 1 and Side 2

We'll begin by applying the triangle inequality theorem to Side 1 and Side 2. Let Side 1 have length 5, Side 2 have length 8, and Side 3 have length c: \(5+8>c\) \(13>c\) Now we know that Side 3 (length c) is less than 13.
02

Apply the triangle inequality theorem to Side 1 and Side 3

Now, let's apply the triangle inequality theorem to Side 1 and Side 3: \( 5+c>8 \) Subtract 5 from both sides of the inequality: \( c>3 \) Now we know that Side 3 (length c) is greater than 3.
03

Apply the triangle inequality theorem to Side 2 and Side 3

Lastly, we'll apply the triangle inequality theorem to Side 2 and Side 3: \( 8+c>5 \) Subtract 8 from both sides of the inequality: \( c>-3 \) However, we had already found that c > 3 in Step 2, so this inequality does not provide any additional useful information.
04

Combine the inequalities

Finally, we can combine the inequalities found in Steps 1 and 2 to find the range for the length of Side 3: \( 3<c<13 \) Thus, the length of the third side of the triangle (c) must be greater than 3 and less than 13.

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