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The length of the sides of a triangle are x + 1, 9 - x and 5x -3 . The number of values of x for which the triangle is isosceles is:
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The length of the sides of a triangle are x + 1, 9 - x and 5x -3 . Th...
In isosceles triangle, 2 sides will be equal
Let x + 1 = 9 - x
2x = 8
X = 4
The sum of any two sides of triangle should be greater than third side
Hence, if x = 4, then sides of the triangle are 5, 5 and 17
This is invalid as the sum of (5+5) is less than 17
Let x + 1 = 5x - 3
X = 1
If x = 1, then the sides of the triangle are 2, 8, 2
This is also invalid
Let 9 - x = 5x - 3
X = 2
If x =2, then the sides of the triangle are 3, 7, 7
Hence, this is valid.
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The length of the sides of a triangle are x + 1, 9 - x and 5x -3 . Th...
Understanding the Triangle's Sides
The sides of the triangle are given as:
- Side 1: x + 1
- Side 2: 9 - x
- Side 3: 5x - 3
To determine when the triangle is isosceles, at least two sides must be equal.
Case 1: x + 1 = 9 - x
- Rearranging gives:
- 2x = 8
- x = 4
Case 2: x + 1 = 5x - 3
- Rearranging gives:
- 4 = 4x
- x = 1
Case 3: 9 - x = 5x - 3
- Rearranging gives:
- 12 = 6x
- x = 2
Summary of Solutions
We have found three potential values for x:
- x = 4
- x = 1
- x = 2
Checking for Validity
Next, we need to confirm that these values satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
1. For x = 4:
- Sides: 5, 5, 17 (not a triangle)
2. For x = 1:
- Sides: 2, 8, 2 (valid triangle)
3. For x = 2:
- Sides: 3, 7, 7 (valid triangle)
Conclusion
After verifying, only x = 1 and x = 2 result in valid triangles. Thus, the number of values of x for which the triangle is isosceles is indeed 1 (option B).
The sides of the triangle are given as:
- Side 1: x + 1
- Side 2: 9 - x
- Side 3: 5x - 3
To determine when the triangle is isosceles, at least two sides must be equal.
Case 1: x + 1 = 9 - x
- Rearranging gives:
- 2x = 8
- x = 4
Case 2: x + 1 = 5x - 3
- Rearranging gives:
- 4 = 4x
- x = 1
Case 3: 9 - x = 5x - 3
- Rearranging gives:
- 12 = 6x
- x = 2
Summary of Solutions
We have found three potential values for x:
- x = 4
- x = 1
- x = 2
Checking for Validity
Next, we need to confirm that these values satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
1. For x = 4:
- Sides: 5, 5, 17 (not a triangle)
2. For x = 1:
- Sides: 2, 8, 2 (valid triangle)
3. For x = 2:
- Sides: 3, 7, 7 (valid triangle)
Conclusion
After verifying, only x = 1 and x = 2 result in valid triangles. Thus, the number of values of x for which the triangle is isosceles is indeed 1 (option B).
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The length of the sides of a triangle are x + 1, 9 - x and 5x -3 . The number of values of x for which the triangle is isosceles is:a)0b)1c)2d)3Correct answer is option 'B'. Can you explain this answer?
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