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The lengths of three sides of a triangle are given by (x + 7), (9 - 2x) and (3x - 1). For how many integral values of x will the triangle be isosceles?
Correct answer is '2'. Can you explain this answer?
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The lengths of three sides of a triangle are given by (x + 7), (9 - 2x...
Solution: An isosceles triangle has two sides equal. We have three cases:
Case (i): x + 7 = 9 - 2x
3x = 2. x = 2/3, which is not an integer.
Case (ii): x + 7 = 3x - 1
2x = 8, x = 4
The sides are 11,11 and 1
Case (iii): 9 - 2x = 3x - 1, 5x = 10, x = 2 The sides are 5, 5 and 9
The triangle is isosceles for two values of x.
Answer: 2
Most Upvoted Answer
The lengths of three sides of a triangle are given by (x + 7), (9 - 2x...
Given:
The lengths of three sides of a triangle are given by (x + 7), (9 - 2x), and (3x - 1).
To find:
The number of integral values of x for which the triangle is isosceles.
Explanation:
A triangle is said to be isosceles if at least two sides of the triangle are equal in length.
In this case, we need to find the values of x for which two sides of the triangle are equal.
Step 1:
Set up the equation to represent the condition for an isosceles triangle.
(x + 7) = (9 - 2x) or (9 - 2x) = (3x - 1) or (3x - 1) = (x + 7)
Step 2:
Solve each equation separately to find the values of x.
For the first equation,
x + 7 = 9 - 2x
3x = 2
x = 2/3
For the second equation,
9 - 2x = 3x - 1
5x = 10
x = 2
For the third equation,
3x - 1 = x + 7
2x = 8
x = 4
Step 3:
Check if the values of x satisfy the conditions for an isosceles triangle.
For x = 2/3, the lengths of the sides are (x + 7) = 19/3, (9 - 2x) = 17/3, and (3x - 1) = 5/3. Since none of the sides are equal, this value of x does not satisfy the condition for an isosceles triangle.
For x = 2, the lengths of the sides are (x + 7) = 9, (9 - 2x) = 5, and (3x - 1) = 5. The triangle formed by these side lengths is isosceles.
For x = 4, the lengths of the sides are (x + 7) = 11, (9 - 2x) = 1, and (3x - 1) = 11. The triangle formed by these side lengths is isosceles.
Conclusion:
Out of the three values of x, only two values (x = 2 and x = 4) satisfy the condition for an isosceles triangle. Therefore, the number of integral values of x for which the triangle is isosceles is 2.
The lengths of three sides of a triangle are given by (x + 7), (9 - 2x), and (3x - 1).
To find:
The number of integral values of x for which the triangle is isosceles.
Explanation:
A triangle is said to be isosceles if at least two sides of the triangle are equal in length.
In this case, we need to find the values of x for which two sides of the triangle are equal.
Step 1:
Set up the equation to represent the condition for an isosceles triangle.
(x + 7) = (9 - 2x) or (9 - 2x) = (3x - 1) or (3x - 1) = (x + 7)
Step 2:
Solve each equation separately to find the values of x.
For the first equation,
x + 7 = 9 - 2x
3x = 2
x = 2/3
For the second equation,
9 - 2x = 3x - 1
5x = 10
x = 2
For the third equation,
3x - 1 = x + 7
2x = 8
x = 4
Step 3:
Check if the values of x satisfy the conditions for an isosceles triangle.
For x = 2/3, the lengths of the sides are (x + 7) = 19/3, (9 - 2x) = 17/3, and (3x - 1) = 5/3. Since none of the sides are equal, this value of x does not satisfy the condition for an isosceles triangle.
For x = 2, the lengths of the sides are (x + 7) = 9, (9 - 2x) = 5, and (3x - 1) = 5. The triangle formed by these side lengths is isosceles.
For x = 4, the lengths of the sides are (x + 7) = 11, (9 - 2x) = 1, and (3x - 1) = 11. The triangle formed by these side lengths is isosceles.
Conclusion:
Out of the three values of x, only two values (x = 2 and x = 4) satisfy the condition for an isosceles triangle. Therefore, the number of integral values of x for which the triangle is isosceles is 2.
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