triangle inequality theorem
state if the three numbers can be the measures of the sides of a triangle.
1) 8, 8, 12
2) 9, 11, 6
3) 8, 3, 11
4) 6, 14, 7
5) 6, 11, 16
two sides of a triangle have the following measures. find the range of possible measures for
the third side.
6) 9, 12
7) 7, 7
8) 10, 8
9) 7, 11
10) 7, 12

options: yes, no, 4

Question

triangle inequality theorem
state if the three numbers can be the measures of the sides of a triangle.
1) 8, 8, 12
2) 9, 11, 6
3) 8, 3, 11
4) 6, 14, 7
5) 6, 11, 16
two sides of a triangle have the following measures. find the range of possible measures for
the third side.
6) 9, 12
7) 7, 7
8) 10, 8
9) 7, 11
10) 7, 12

options: yes, no, 4 < x < 18, 2 < x < 18, 3 < x < 21, 5 < x < 19, 0 < x < 14

Accepted Answer

To solve these triangle inequality problems, we use the **Triangle Inequality Theorem**, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For a triangle with sides $ a $, $ b $, and $ c $, the three conditions are: $ a + b > c $, $ a + c > b $, and $ b + c > a $. ### Problem 1: $ 8, 8, 12 $ Check all three inequalities: - $ 8 + 8 > 12 $: $ 16 > 12 $ (True) - $ 8 + 12 > 8 $: $ 20 > 8 $ (True) - $ 8 + 12 > 8 $: $ 20 > 8 $ (True) All conditions are satisfied. ### Problem 2: $ 9, 11, 6 $ Check all three inequalities: - $ 9 + 11 > 6 $: $ 20 > 6 $ (True) - $ 9 + 6 > 11 $: $ 15 > 11 $ (True) - $ 11 + 6 > 9 $: $ 17 > 9 $ (True) All conditions are satisfied. ### Problem 3: $ 8, 3, 11 $ Check the sum of the two smaller sides: $ 8 + 3 = 11 $. The theorem requires the sum to be **greater than** the third side, not equal. Thus: $ 8 + 3 ot> 11 $ (False). ### Problem 4: $ 6, 14, 7 $ Check the sum of the two smaller sides: $ 6 + 7 = 13 $. Compare to the largest side ($ 14 $): $ 6 + 7 ot> 14 $ (False). ### Problem 5: $ 6, 11, 16 $ Check the sum of the two smaller sides: $ 6 + 11 = 17 $. Compare to the largest side ($ 16 $): $ 6 + 11 > 16 $ (True). Now check the other inequalities: - $ 6 + 16 > 11 $: $ 22 > 11 $ (True) - $ 11 + 16 > 6 $: $ 27 > 6 $ (True) All conditions are satisfied. ### Finding the Range of the Third Side For a triangle with two sides $ a $ and $ b $ (where $ a \geq b $), the length of the third side $ x $ must satisfy: $ a - b < x < a + b $. ### Problem 6: Two sides $ 3, 12 $ Let $ a = 12 $, $ b = 3 $. Then: $ 12 - 3 < x < 12 + 3 $ $ 9 < x < 15 $? Wait, no—wait, the given options include $ 4 < x < 18 $, $ 2 < x < 18 $, $ 3 < x < 21 $, $ 5 < x < 19 $, $ 0 < x < 14 $. Wait, maybe I misread the problem. Let’s re-express: For two sides $ a $ and $ b $, the third side $ x $ must satisfy: $ |a - b| < x < a + b $. #### Problem 6: Sides $ 3, 12 $ $ |12 - 3| = 9 $, $ 12 + 3 = 15 $. But the given options don’t have $ 9 < x < 15 $. Wait, maybe the sides are $ 10, 8 $? No, problem 6 is labeled $ 3, 12 $? Wait, the user’s image shows: - 6) Two sides: $ 3, 12 $ - 7) Two sides: $ 7, 7 $ - 8) Two sides: $ 10, 8 $ - 9) Two sides: $ 7, 11 $ - 10) Two sides: $ 7, 12 $ Let’s solve each: ### Problem 6: Sides $ 3, 12 $ $ |12 - 3| = 9 $, $ 12 + 3 = 15 $. No matching option? Wait, the given options are: $ 4 < x < 18 $, $ 2 < x < 18 $, $ 3 < x < 21 $, $ 5 < x < 19 $, $ 0 < x < 14 $. Maybe a typo, but let’s check the other problems. ### Problem 7: Sides $ 7, 7 $ $ |7 - 7| = 0 $, $ 7 + 7 = 14 $. So $ 0 < x < 14 $. ### Problem 8: Sides $ 10, 8 $ $ |10 - 8| = 2 $, $ 10 + 8 = 18 $. So $ 2 < x < 18 $. ### Problem 9: Sides $ 7, 11 $ $ |11 - 7| = 4 $, $ 11 + 7 = 18 $. So $ 4 < x < 18 $. ### Problem 10: Sides $ 7, 12 $ $ |12 - 7| = 5 $, $ 12 + 7 = 19 $. So $ 5 < x < 19 $. ### Final Answers (Matching the Options) 1) $ 8, 8, 12 $: Yes 2) $ 9, 11, 6 $: Yes 3) $ 8, 3, 11 $: No 4) $ 6, 14, 7 $: No 5) $ 6, 11, 16 $: Yes 6) $ 3, 12 $: Wait, no matching option? Wait, maybe the sides are $ 10, 8 $? No, problem 6 is $ 3, 12 $. Alternatively, maybe the first side is $ 10 $, not $ 3 $? Let’s assume the user’s image has: - 6) Sides $ 10, 8 $? No, problem 6 is labeled $ 3, 12 $. Alternatively, the options are: Given the options: - $ 4 < x < 18 $: Matches problem 9 ($ 7, 11 $) - $ 2 < x < 18 $: Matches problem 8 ($ 10, 8 $) - $ 0 < x < 14 $: Matches problem 7 ($ 7, 7 $) - $ 5 < x < 19 $: Matches problem 10 ($ 7, 12 $) ### Summary of Answers: 1) Yes 2) Yes 3) No 4) No 5) Yes 6) (Sides $ 3, 12 $): $ 9 < x < 15 $ (not in options, but likely a typo) 7) $ 0 < x < 14 $ 8) $ 2 < x < 18 $ 9) $ 4 < x < 18 $ 10) $ 5 < x < 19 $ ### Final Answers (Matching the Image’s Options): 1) Yes 2) Yes 3) No 4) No 5) Yes 6) (Assuming typo, but if forced to choose from given options, none match. Wait, maybe the first side is $ 10 $, not $ 3 $. Let’s re-express: If problem 6 is $ 10, 8 $: $ 2 < x < 18 $ (option 2 < x < 18). But based on the image, the answers are: 1) Yes 2) Yes 3) No 4) No 5) Yes 6) $ 9 < x < 15 $ (not in options, but maybe the user’s options are mislabeled) 7) $ 0 < x < 14 $ 8) $ 2 < x < 18 $ 9) $ 4 < x < 18 $ 10) $ 5 < x < 19 $ For the first five (side length checks): 1) Yes 2) Yes 3) No 4) No 5) Yes For the range problems: 6) (Sides $ 3, 12 $): $ 9 < x < 15 $ (no option, but likely error) 7) $ 0 < x < 14 $ 8) $ 2 < x < 18 $ 9) $ 4 < x < 18 $ 10) $ 5 < x < 19 $ If we must use the given options: - Problem 7: $ 0 < x < 14 $ (matches) - Problem 8: $ 2 < x < 18 $ (matches) - Problem 9: $ 4 < x < 18 $ (matches) - Problem 10: $ 5 < x < 19 $ (matches) ### Final Answers (Step-by-Step for Side Checks): #### 1) $ 8, 8, 12 $ - $ 8 + 8 > 12 $: $ 16 > 12 $ (True) - $ 8 + 12 > 8 $: $ 20 > 8 $ (True) - $ 8 + 12 > 8 $: $ 20 > 8 $ (True) Answer: Yes #### 2) $ 9, 11, 6 $ - $ 9 + 11 > 6 $: $ 20 > 6 $ (True) - $ 9 + 6 > 11 $: $ 15 > 11 $ (True) - $ 11 + 6 > 9 $: $ 17 > 9 $ (True) Answer: Yes #### 3) $ 8, 3, 11 $ - $ 8 + 3 = 11 $, but $ 8 + 3 ot> 11 $ (False) Answer: No #### 4) $ 6, 14, 7 $ - $ 6 + 7 = 13 $, and $ 13 ot> 14 $ (False) Answer: No #### 5) $ 6, 11, 16 $ - $ 6 + 11 = 17 > 16 $ (True) - $ 6 + 16 > 11 $: $ 22 > 11 $ (True) - $ 11 + 16 > 6 $: $ 27 > 6 $ (True) Answer: Yes ### Range Problems: #### 7) $ 7, 7 $ - $ |7 - 7| = 0 $, $ 7 + 7 = 14 $ - Range: $ 0 < x < 14 $ Answer: $ 0 < x < 14 $ #### 8) $ 10, 8 $ - $ |10 - 8| = 2 $, $ 10 + 8 = 18 $ - Range: $ 2 < x < 18 $ Answer: $ 2 < x < 18 $ #### 9) $ 7, 11 $ - $ |11 - 7| = 4 $, $ 11 + 7 = 18 $ - Range: $ 4 < x < 18 $ Answer: $ 4 < x < 18 $ #### 10) $ 7, 12 $ - $ |12 - 7| = 5 $, $ 12 + 7 = 19 $ - Range: $ 5 < x < 19 $ Answer: $ 5 < x < 19 $ (Problem 6: Likely a typo, but if we assume sides $ 3, 12 $, the range is $ 9 < x < 15 $, not in the given options.)

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( 8, 8, 12 \)

Problem 2: \( 9, 11, 6 \)

Problem 3: \( 8, 3, 11 \)

Problem 4: \( 6, 14, 7 \)

Problem 5: \( 6, 11, 16 \)

Finding the Range of the Third Side

Problem 6: Two sides \( 3, 12 \)

Problem 6: Sides \( 3, 12 \)

Problem 6: Sides \( 3, 12 \)

Problem 7: Sides \( 7, 7 \)

Problem 8: Sides \( 10, 8 \)

Problem 9: Sides \( 7, 11 \)

Problem 10: Sides \( 7, 12 \)

Final Answers (Matching the Options)

Answer:

2) \( 9, 11, 6 \)