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How many differently shaped triangles exist in which no two sides are of the same length, each side is of integral unit length and the perimeter of the triangle is less than 14 units?
- a)3
- b)4
- c)5
- d)6
- e)None of the above
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
How many differently shaped triangles exist in which no two sides are ...
Since the perimeter of the triangle is less than 14 units and all the sides are of integral lengths we can have the following triangles
(1) (2, 3, 4)
(2) (2, 4, 5)
(3) (2, 5, 6)
(4) (3, 4, 5)
(5) (3, 4, 6)
Since no two sides of the triangle must be same and in a triangle sum of two sides is more than the third side, these are the only possible cases.
(1) (2, 3, 4)
(2) (2, 4, 5)
(3) (2, 5, 6)
(4) (3, 4, 5)
(5) (3, 4, 6)
Since no two sides of the triangle must be same and in a triangle sum of two sides is more than the third side, these are the only possible cases.
Most Upvoted Answer
How many differently shaped triangles exist in which no two sides are ...
To solve this problem, we need to find all the possible triangles that satisfy the given conditions. Let's analyze the conditions and possible values for each side of the triangle.
1. No two sides are of the same length: This means that all three sides of the triangle must have different lengths.
2. Each side is of integral unit length: This implies that the lengths of the sides can only be positive integers.
3. The perimeter of the triangle is less than 14 units: The sum of the lengths of all three sides must be less than 14.
Now, let's consider the possible lengths for each side of the triangle. Since the sides must be of different lengths, we have the following options:
Side 1: 1 unit
Side 2: 2 units
Side 3: 3 units
Side 1: 1 unit
Side 2: 2 units
Side 3: 4 units
Side 1: 1 unit
Side 2: 2 units
Side 3: 5 units
Side 1: 1 unit
Side 2: 3 units
Side 3: 4 units
Side 1: 1 unit
Side 2: 3 units
Side 3: 5 units
Side 1: 2 units
Side 2: 3 units
Side 3: 4 units
Side 1: 2 units
Side 2: 3 units
Side 3: 5 units
Side 1: 2 units
Side 2: 4 units
Side 3: 5 units
Side 1: 3 units
Side 2: 4 units
Side 3: 5 units
Now, let's calculate the perimeter of each triangle:
Triangle 1: 1 + 2 + 3 = 6 units
Triangle 2: 1 + 2 + 4 = 7 units
Triangle 3: 1 + 2 + 5 = 8 units
Triangle 4: 1 + 3 + 4 = 8 units
Triangle 5: 1 + 3 + 5 = 9 units
Triangle 6: 2 + 3 + 4 = 9 units
Triangle 7: 2 + 3 + 5 = 10 units
Triangle 8: 2 + 4 + 5 = 11 units
Triangle 9: 3 + 4 + 5 = 12 units
From the above calculations, we see that there are 5 triangles whose perimeters are less than 14 units:
- Triangle 1: 1-2-3
- Triangle 2: 1-2-4
- Triangle 3: 1-2-5
- Triangle 4: 1-3-4
- Triangle 5: 1-3-5
Therefore, the correct answer is option C) 5.
1. No two sides are of the same length: This means that all three sides of the triangle must have different lengths.
2. Each side is of integral unit length: This implies that the lengths of the sides can only be positive integers.
3. The perimeter of the triangle is less than 14 units: The sum of the lengths of all three sides must be less than 14.
Now, let's consider the possible lengths for each side of the triangle. Since the sides must be of different lengths, we have the following options:
Side 1: 1 unit
Side 2: 2 units
Side 3: 3 units
Side 1: 1 unit
Side 2: 2 units
Side 3: 4 units
Side 1: 1 unit
Side 2: 2 units
Side 3: 5 units
Side 1: 1 unit
Side 2: 3 units
Side 3: 4 units
Side 1: 1 unit
Side 2: 3 units
Side 3: 5 units
Side 1: 2 units
Side 2: 3 units
Side 3: 4 units
Side 1: 2 units
Side 2: 3 units
Side 3: 5 units
Side 1: 2 units
Side 2: 4 units
Side 3: 5 units
Side 1: 3 units
Side 2: 4 units
Side 3: 5 units
Now, let's calculate the perimeter of each triangle:
Triangle 1: 1 + 2 + 3 = 6 units
Triangle 2: 1 + 2 + 4 = 7 units
Triangle 3: 1 + 2 + 5 = 8 units
Triangle 4: 1 + 3 + 4 = 8 units
Triangle 5: 1 + 3 + 5 = 9 units
Triangle 6: 2 + 3 + 4 = 9 units
Triangle 7: 2 + 3 + 5 = 10 units
Triangle 8: 2 + 4 + 5 = 11 units
Triangle 9: 3 + 4 + 5 = 12 units
From the above calculations, we see that there are 5 triangles whose perimeters are less than 14 units:
- Triangle 1: 1-2-3
- Triangle 2: 1-2-4
- Triangle 3: 1-2-5
- Triangle 4: 1-3-4
- Triangle 5: 1-3-5
Therefore, the correct answer is option C) 5.
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Community Answer
How many differently shaped triangles exist in which no two sides are ...
Since the perimeter of the triangle is less than 14 units and all the sides are of integral lengths we can have the following triangles
(1) (2, 3, 4)
(2) (2, 4, 5)
(3) (2, 5, 6)
(4) (3, 4, 5)
(5) (3, 4, 6)
Since no two sides of the triangle must be same and in a triangle sum of two sides is more than the third side, these are the only possible cases.
(1) (2, 3, 4)
(2) (2, 4, 5)
(3) (2, 5, 6)
(4) (3, 4, 5)
(5) (3, 4, 6)
Since no two sides of the triangle must be same and in a triangle sum of two sides is more than the third side, these are the only possible cases.
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How many differently shaped triangles exist in which no two sides are of the same length, each side is of integral unit length and the perimeter of the triangle is less than 14 units?a)3b)4c)5d)6e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?
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