In Exercises 15-22, determine whether the segment
lengths form a triangle. If so, is the triangle acute, right,
or obtuse? (See Example 5.)
15. 10,11 , and 14
16. 6,8 , and 14
17. 12,16 , and 20
18. 15,20 , and 26
19. , and 12.4
20. , and 7.8
21. 24,30 , and
22. 10,15 , and

Ask by Harris Burgess. in the United States

Mar 25,2025

Question

In Exercises 15-22, determine whether the segment
lengths form a triangle. If so, is the triangle acute, right,
or obtuse? (See Example 5.)
15. 10,11 , and 14
16. 6,8 , and 14
17. 12,16 , and 20
18. 15,20 , and 26
19. , and 12.4
20. , and 7.8
21. 24,30 , and
22. 10,15 , and

Ask by Harris Burgess. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

To determine whether the given segment lengths can form a triangle, we will use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths $$ a $$, $$ b $$, and $$ c $$: 1. $$ a + b > c $$ 2. $$ a + c > b $$ 3. $$ b + c > a $$ If the lengths satisfy all three conditions, they can form a triangle. Once we establish that a triangle can be formed, we will determine whether it is acute, right, or obtuse by using the following criteria based on the squares of the lengths: - If $$ a^2 + b^2 > c^2 $$, the triangle is acute. - If $$ a^2 + b^2 = c^2 $$, the triangle is right. - If $$ a^2 + b^2 < c^2 $$, the triangle is obtuse. Let's analyze each set of segment lengths step by step. ### 15. $$ 10, 11, 14 $$ 1. Check triangle inequality: - $$ 10 + 11 > 14 $$ (True) - $$ 10 + 14 > 11 $$ (True) - $$ 11 + 14 > 10 $$ (True) Since all conditions are satisfied, a triangle can be formed. 2. Determine the type: - $$ 10^2 + 11^2 = 100 + 121 = 221 $$ - $$ 14^2 = 196 $$ - $$ 221 > 196 $$ (Acute) ### 16. $$ 6, 8, 14 $$ 1. Check triangle inequality: - $$ 6 + 8 > 14 $$ (False) A triangle cannot be formed. ### 17. $$ 12, 16, 20 $$ 1. Check triangle inequality: - $$ 12 + 16 > 20 $$ (True) - $$ 12 + 20 > 16 $$ (True) - $$ 16 + 20 > 12 $$ (True) A triangle can be formed. 2. Determine the type: - $$ 12^2 + 16^2 = 144 + 256 = 400 $$ - $$ 20^2 = 400 $$ - $$ 400 = 400 $$ (Right) ### 18. $$ 15, 20, 26 $$ 1. Check triangle inequality: - $$ 15 + 20 > 26 $$ (True) - $$ 15 + 26 > 20 $$ (True) - $$ 20 + 26 > 15 $$ (True) A triangle can be formed. 2. Determine the type: - $$ 15^2 + 20^2 = 225 + 400 = 625 $$ - $$ 26^2 = 676 $$ - $$ 625 < 676 $$ (Obtuse) ### 19. $$ 4.1, 8.2, 12.4 $$ 1. Check triangle inequality: - $$ 4.1 + 8.2 > 12.4 $$ (False) A triangle cannot be formed. ### 20. $$ 5.3, 6.7, 7.8 $$ 1. Check triangle inequality: - $$ 5.3 + 6.7 > 7.8 $$ (True) - $$ 5.3 + 7.8 > 6.7 $$ (True) - $$ 6.7 + 7.8 > 5.3 $$ (True) A triangle can be formed. 2. Determine the type: - $$ 5.3^2 + 6.7^2 = 28.09 + 44.89 = 72.98 $$ - $$ 7.8^2 = 60.84 $$ - $$ 72.98 > 60.84 $$ (Acute) ### 21. $$ 24, 30, 6\sqrt{43} $$ 1. Check triangle inequality: - $$ 24 + 30 > 6\sqrt{43} $$ (True) - $$ 24 + 6\sqrt{43} > 30 $$ (True) - $$ 30 + 6\sqrt{43} > 24 $$ (True) A triangle can be formed. 2. Determine the type: - $$ 24^2 + 30^2 = 576 + 900 = 1476 $$ - $$ (6\sqrt{43})^2 = 36 imes 43 = 1548 $$ - $$ 1476 < 1548 $$ (Obtuse) ### 22. $$ 10, 15, 5\sqrt{13} $$ 1. Check triangle inequality: - $$ 10 + 15 > 5\sqrt{13} $$ (True) - $$ 10 + 5\sqrt{13} > 15 $$ (True) - $$ 15 + 5\sqrt{13} > 10 $$ (True) A triangle can be formed. 2. Determine the type: - $$ 10^2 + 15^2 = 100 + 225 = 325 $$ - $$ (5\sqrt{13})^2 = 25 imes 13 = 325 $$ - $$ 325 = 325 $$ (Right) ### Summary of Results: - **15:** Triangle (Acute) - **16:** No triangle - **17:** Triangle (Right) - **18:** Triangle (Obtuse) - **19:** No triangle - **20:** Triangle (Acute) - **21:** Triangle (Obtuse) - **22:** Triangle (Right) - **15:** Triangle (Acute) - **16:** No triangle - **17:** Triangle (Right) - **18:** Triangle (Obtuse) - **19:** No triangle - **20:** Triangle (Acute) - **21:** Triangle (Obtuse) - **22:** Triangle (Right)

In Exercises 15-22, determine whether the segment
lengths form a triangle. If so, is the triangle acute, right,
or obtuse? (See Example 5.)
15. 10,11 , and 14
16. 6,8 , and 14
17. 12,16 , and 20
18. 15,20 , and 26
19. , and 12.4
20. , and 7.8
21. 24,30 , and
22. 10,15 , and

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In Exercises 15-22, determine whether the segment lengths form a triangle. If so, is the triangle acute, right, or obtuse? (See Example 5.) 15. 10,11 , and 14 16. 6,8 , and 14 17. 12,16 , and 20 18. 15,20 , and 26 19. , and 12.4 20. , and 7.8 21. 24,30 , and 22. 10,15 , and

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Latest Geometry Questions

In Exercises 15-22, determine whether the segment
lengths form a triangle. If so, is the triangle acute, right,
or obtuse? (See Example 5.)
15. 10,11 , and 14
16. 6,8 , and 14
17. 12,16 , and 20
18. 15,20 , and 26
19. , and 12.4
20. , and 7.8
21. 24,30 , and
22. 10,15 , and