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The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is
- a)5
- b)4
- c)3
- d)6
Correct answer is option 'A'. Can you explain this answer?
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The lengths of all four sides of a quadrilateral are integer valued. I...
To find the possible lengths of the fourth side of the quadrilateral, we need to consider the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's consider the given lengths of the three sides: 1 cm, 2 cm, and 4 cm. Now, we need to find the possible lengths of the fourth side that satisfy the triangle inequality theorem.
1. Case 1: The fourth side is smaller than 1 cm
If the fourth side is smaller than 1 cm, it is not possible to form a triangle because the sum of any two sides will always be greater than the length of the third side. Therefore, we can eliminate this case.
2. Case 2: The fourth side is equal to 1 cm
If the fourth side is equal to 1 cm, we can form a triangle with the given side lengths. Therefore, 1 cm is a possible length for the fourth side.
3. Case 3: The fourth side is between 1 cm and 2 cm
If the fourth side is between 1 cm and 2 cm, we can form a triangle with the given side lengths. Therefore, any length between 1 cm and 2 cm (excluding 1 cm and 2 cm) is a possible length for the fourth side.
4. Case 4: The fourth side is equal to 2 cm
If the fourth side is equal to 2 cm, we can form a triangle with the given side lengths. Therefore, 2 cm is a possible length for the fourth side.
5. Case 5: The fourth side is between 2 cm and 4 cm
If the fourth side is between 2 cm and 4 cm, we can form a triangle with the given side lengths. Therefore, any length between 2 cm and 4 cm (excluding 2 cm and 4 cm) is a possible length for the fourth side.
6. Case 6: The fourth side is equal to 4 cm
If the fourth side is equal to 4 cm, it is not possible to form a triangle because the sum of the lengths of the three given sides (1 cm + 2 cm + 4 cm) is equal to the length of the fourth side. Therefore, we can eliminate this case.
In total, we have 5 possible lengths for the fourth side: 1 cm, any length between 1 cm and 2 cm, 2 cm, and any length between 2 cm and 4 cm. Therefore, the correct answer is option A) 5.
Let's consider the given lengths of the three sides: 1 cm, 2 cm, and 4 cm. Now, we need to find the possible lengths of the fourth side that satisfy the triangle inequality theorem.
1. Case 1: The fourth side is smaller than 1 cm
If the fourth side is smaller than 1 cm, it is not possible to form a triangle because the sum of any two sides will always be greater than the length of the third side. Therefore, we can eliminate this case.
2. Case 2: The fourth side is equal to 1 cm
If the fourth side is equal to 1 cm, we can form a triangle with the given side lengths. Therefore, 1 cm is a possible length for the fourth side.
3. Case 3: The fourth side is between 1 cm and 2 cm
If the fourth side is between 1 cm and 2 cm, we can form a triangle with the given side lengths. Therefore, any length between 1 cm and 2 cm (excluding 1 cm and 2 cm) is a possible length for the fourth side.
4. Case 4: The fourth side is equal to 2 cm
If the fourth side is equal to 2 cm, we can form a triangle with the given side lengths. Therefore, 2 cm is a possible length for the fourth side.
5. Case 5: The fourth side is between 2 cm and 4 cm
If the fourth side is between 2 cm and 4 cm, we can form a triangle with the given side lengths. Therefore, any length between 2 cm and 4 cm (excluding 2 cm and 4 cm) is a possible length for the fourth side.
6. Case 6: The fourth side is equal to 4 cm
If the fourth side is equal to 4 cm, it is not possible to form a triangle because the sum of the lengths of the three given sides (1 cm + 2 cm + 4 cm) is equal to the length of the fourth side. Therefore, we can eliminate this case.
In total, we have 5 possible lengths for the fourth side: 1 cm, any length between 1 cm and 2 cm, 2 cm, and any length between 2 cm and 4 cm. Therefore, the correct answer is option A) 5.
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The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side isa)5b)4c)3d)6Correct answer is option 'A'. Can you explain this answer?
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