QUESTION IMAGE
Question
directions: determine if the side lengths could form a triangle. use an inequality to prove your answer.
- 16 m, 11 m, 30 m
- 10 ft, 8 ft, 14 ft
- 34 km, 12 km, 80 km
- 29 ft, 30 ft, 9 ft
- 12 cm, 12 cm, 25 cm
- 31 yd, 14 yd, 19 yd
directions: given two sides of a triangle, find a range of possible lengths for the third side.
- 4 cm, 17 cm
- 24 ft, 52 ft
- 9 yd, 32 yd
- 16 km, 17 km
- if a triangle has lengths of 27 m and 11 m, check all the possible lengths for the third side.
□ 39 ft □ 17 ft □ 35 ft □ 22 ft □ 16 ft
- if a triangle has lengths of 3 ft and 54 ft, check all the possible lengths for the third side.
□ 51 ft □ 53 ft □ 55 ft □ 57 ft □ 58 ft
directions: order the angles from least to greatest for 13-14, then greatest to least for 15-16.
- triangle ( nqr ) with sides 21 m, 24 m, 17 m
- triangle ( gjh ) with sides 6 in, 5 in, 11 in
To solve these triangle - related problems, we will use the Triangle Inequality Theorem. The theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\):
- \(a + b>c\)
- \(a + c>b\)
- \(b + c>a\)
For finding the range of the third side when two sides (\(a\) and \(b\)) are given, the length of the third side \(x\) must satisfy \(|a - b|\lt x\lt a + b\)
Problem 1: Determine if side lengths form a triangle (e.g., 16 m, 11 m, 30 m)
Step 1: Identify the sides
Let \(a = 16\), \(b = 11\), \(c = 30\)
Step 2: Check the triangle inequalities
- \(a + b=16 + 11 = 27\). Since \(27<30\) (i.e., \(a + b
ot>c\)), the side lengths \(16\) m, \(11\) m, and \(30\) m cannot form a triangle.
Problem 7: Given two sides (4 cm, 17 cm), find the range of the third side
Step 1: Apply the range formula
Let \(a = 4\) and \(b = 17\). The length of the third side \(x\) must satisfy \(|a - b|\lt x\lt a + b\)
Step 2: Calculate the bounds
- \(|4 - 17|=| - 13| = 13\)
- \(4+17 = 21\)
So the range of the third side is \(13\space\text{cm}\lt x\lt21\space\text{cm}\)
Problem 11: Given two sides (27 m, 11 m), check possible third - side lengths
Step 1: Find the range of the third side
Using the formula \(|a - b|\lt x\lt a + b\), where \(a = 27\) and \(b = 11\)
- \(|27 - 11|=16\)
- \(27 + 11 = 38\)
So the third side \(x\) must satisfy \(16\lt x\lt38\)
Step 2: Check each option
- \(39\) ft: \(39>38\), so it is not valid.
- \(17\) ft: \(16 < 17<38\), valid.
- \(35\) ft: \(16 < 35<38\), valid.
- \(22\) ft: \(16 < 22<38\), valid.
- \(16\) ft: \(16\) is not greater than \(16\) (the lower bound is strict), so it is not valid.
Problem 13: Order the angles of \(\triangle NRQ\) with sides \(NR = 24\) m, \(RQ = 17\) m, \(QN = 21\) m
In a triangle, the larger the side length, the larger the angle opposite to it.
- The side lengths are \(17\) m (\(RQ\)), \(21\) m (\(QN\)), and \(24\) m (\(NR\))
- The angles opposite to these sides are \(\angle N\), \(\angle R\), and \(\angle Q\) respectively.
Since \(17<21<24\), the angles from least to greatest are \(\angle N<\angle Q<\angle R\)
Problem 14: Order the angles of \(\triangle GJH\) with sides \(GJ = 6\) in, \(JH = 5\) in, \(GH = 11\) in
- The side lengths are \(5\) in (\(JH\)), \(6\) in (\(GJ\)), and \(11\) in (\(GH\))
- The angles opposite to these sides are \(\angle G\), \(\angle H\), and \(\angle J\) respectively.
Since \(5 < 6<11\), the angles from least to greatest are \(\angle G<\angle H<\angle J\)
Final Answers (for specific sub - problems)
- For the problem of whether \(16\) m, \(11\) m, \(30\) m form a triangle: No
- For the range of the third side when two sides are \(4\) cm and \(17\) cm: \(13\space\text{cm}\lt x\lt21\space\text{cm}\)
- For problem 11 (possible third - side lengths): \(17\) ft, \(35\) ft, \(22\) ft
- For problem 13 (order of angles): \(\angle N<\angle Q<\angle R\)
- For problem 14 (order of angles): \(\angle G<\angle H<\angle J\)
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To solve these triangle - related problems, we will use the Triangle Inequality Theorem. The theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\):
- \(a + b>c\)
- \(a + c>b\)
- \(b + c>a\)
For finding the range of the third side when two sides (\(a\) and \(b\)) are given, the length of the third side \(x\) must satisfy \(|a - b|\lt x\lt a + b\)
Problem 1: Determine if side lengths form a triangle (e.g., 16 m, 11 m, 30 m)
Step 1: Identify the sides
Let \(a = 16\), \(b = 11\), \(c = 30\)
Step 2: Check the triangle inequalities
- \(a + b=16 + 11 = 27\). Since \(27<30\) (i.e., \(a + b
ot>c\)), the side lengths \(16\) m, \(11\) m, and \(30\) m cannot form a triangle.
Problem 7: Given two sides (4 cm, 17 cm), find the range of the third side
Step 1: Apply the range formula
Let \(a = 4\) and \(b = 17\). The length of the third side \(x\) must satisfy \(|a - b|\lt x\lt a + b\)
Step 2: Calculate the bounds
- \(|4 - 17|=| - 13| = 13\)
- \(4+17 = 21\)
So the range of the third side is \(13\space\text{cm}\lt x\lt21\space\text{cm}\)
Problem 11: Given two sides (27 m, 11 m), check possible third - side lengths
Step 1: Find the range of the third side
Using the formula \(|a - b|\lt x\lt a + b\), where \(a = 27\) and \(b = 11\)
- \(|27 - 11|=16\)
- \(27 + 11 = 38\)
So the third side \(x\) must satisfy \(16\lt x\lt38\)
Step 2: Check each option
- \(39\) ft: \(39>38\), so it is not valid.
- \(17\) ft: \(16 < 17<38\), valid.
- \(35\) ft: \(16 < 35<38\), valid.
- \(22\) ft: \(16 < 22<38\), valid.
- \(16\) ft: \(16\) is not greater than \(16\) (the lower bound is strict), so it is not valid.
Problem 13: Order the angles of \(\triangle NRQ\) with sides \(NR = 24\) m, \(RQ = 17\) m, \(QN = 21\) m
In a triangle, the larger the side length, the larger the angle opposite to it.
- The side lengths are \(17\) m (\(RQ\)), \(21\) m (\(QN\)), and \(24\) m (\(NR\))
- The angles opposite to these sides are \(\angle N\), \(\angle R\), and \(\angle Q\) respectively.
Since \(17<21<24\), the angles from least to greatest are \(\angle N<\angle Q<\angle R\)
Problem 14: Order the angles of \(\triangle GJH\) with sides \(GJ = 6\) in, \(JH = 5\) in, \(GH = 11\) in
- The side lengths are \(5\) in (\(JH\)), \(6\) in (\(GJ\)), and \(11\) in (\(GH\))
- The angles opposite to these sides are \(\angle G\), \(\angle H\), and \(\angle J\) respectively.
Since \(5 < 6<11\), the angles from least to greatest are \(\angle G<\angle H<\angle J\)
Final Answers (for specific sub - problems)
- For the problem of whether \(16\) m, \(11\) m, \(30\) m form a triangle: No
- For the range of the third side when two sides are \(4\) cm and \(17\) cm: \(13\space\text{cm}\lt x\lt21\space\text{cm}\)
- For problem 11 (possible third - side lengths): \(17\) ft, \(35\) ft, \(22\) ft
- For problem 13 (order of angles): \(\angle N<\angle Q<\angle R\)
- For problem 14 (order of angles): \(\angle G<\angle H<\angle J\)