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ULTIPLE SELECT: If a triangle has lengths of 3 ft and 54 ft, check all the possible lengths for the third side. O 58 t O 51.1 1 O 50.9 ft O 55,1 t O 51 t O 57 n O 53 t O 55 ft

ULTIPLE SELECT: If a triangle has lengths of 3 ft and 54 ft, check all the possible lengths for the third side. O 58 t O 51.1 1 O 50.9 ft O 55,1 t O 51 t O 57 n O 53 t O 55 ft

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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--- ### Triangle Side Lengths Problem **MULTIPLE SELECT:** If a triangle has lengths of 3 ft and 54 ft, check all the possible lengths for the third side. - [ ] 58 ft - [ ] 51.1 ft - [ ] 50.9 ft - [ ] 55.1 ft - [ ] 51 ft - [ ] 57 ft - [ ] 53 ft - [ ] 55 ft --- **Explanation**: To determine the possible lengths for the third side of a triangle when two sides are already known, we must use the triangle inequality theorem. The theorem 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 this problem: - Let the sides be represented as \(a = 3 \, \text{ft}, b = 54 \, \text{ft}, c\) (the unknown third side). ### Conditions for \(c\): 1. \(a + b > c\) 2. \(a + c > b\) 3. \(b + c > a\) Substituting the known values: 1. \(3 + 54 > c \Rightarrow 57 > c \, \text{or} \, c < 57 \, \text{ft}\) 2. \(3 + c > 54 \Rightarrow c > 51 \, \text{ft}\) 3. \(54 + c > 3 \Rightarrow\) This condition is always true as it simplifies to \(c > -51\). Thus, the length of the third side \(c\) must satisfy \(51 \, \text{ft} < c < 57 \, \text{ft}\). ### Conclusion Therefore, the possible lengths of the third side are: - 51.1 ft - 50.9 ft (this doesn't satisfy the condition \(> 51 \, \text{ft}\)) - 55.1 ft - 51 ft (doesn't satisfy the \(> 51 \, \text{ft}\) strictly condition) - 53 ft - 55 ft So, the correct lengths to check are: - [x] 51.1 ft - [x] 55.1 ft - [x] 53 ft - [x]
Transcribed Image Text:--- ### Triangle Side Lengths Problem **MULTIPLE SELECT:** If a triangle has lengths of 3 ft and 54 ft, check all the possible lengths for the third side. - [ ] 58 ft - [ ] 51.1 ft - [ ] 50.9 ft - [ ] 55.1 ft - [ ] 51 ft - [ ] 57 ft - [ ] 53 ft - [ ] 55 ft --- **Explanation**: To determine the possible lengths for the third side of a triangle when two sides are already known, we must use the triangle inequality theorem. The theorem 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 this problem: - Let the sides be represented as \(a = 3 \, \text{ft}, b = 54 \, \text{ft}, c\) (the unknown third side). ### Conditions for \(c\): 1. \(a + b > c\) 2. \(a + c > b\) 3. \(b + c > a\) Substituting the known values: 1. \(3 + 54 > c \Rightarrow 57 > c \, \text{or} \, c < 57 \, \text{ft}\) 2. \(3 + c > 54 \Rightarrow c > 51 \, \text{ft}\) 3. \(54 + c > 3 \Rightarrow\) This condition is always true as it simplifies to \(c > -51\). Thus, the length of the third side \(c\) must satisfy \(51 \, \text{ft} < c < 57 \, \text{ft}\). ### Conclusion Therefore, the possible lengths of the third side are: - 51.1 ft - 50.9 ft (this doesn't satisfy the condition \(> 51 \, \text{ft}\)) - 55.1 ft - 51 ft (doesn't satisfy the \(> 51 \, \text{ft}\) strictly condition) - 53 ft - 55 ft So, the correct lengths to check are: - [x] 51.1 ft - [x] 55.1 ft - [x] 53 ft - [x]
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