To determine the value of x such that the triangle with sides 3x, 5x, and 12 is isosceles, we need to identify the conditions under which at least two sides of the triangle are equal.
In an isosceles triangle, we can have the following cases:
- 3x = 5x
- 3x = 12
- 5x = 12
We will evaluate each case:
Case 1: 3x = 5x
This leads to:
3x – 5x = 0
-2x = 0
This suggests that x = 0, but this does not provide a valid triangle, as all side lengths must be positive.
Case 2: 3x = 12
Solving for x gives:
3x = 12
x = 12 / 3 = 4
Now let’s check the side lengths:
When x = 4:
- 3x = 3 * 4 = 12
- 5x = 5 * 4 = 20
The sides are 12, 20, 12 — which is an isosceles triangle as two sides are equal.
Case 3: 5x = 12
Solving for x gives:
5x = 12
x = 12 / 5 = 2.4
With x = 2.4:
- 3x = 3 * 2.4 = 7.2
- 5x = 5 * 2.4 = 12
The sides are 7.2, 12, 7.2 — again forming an isosceles triangle.
Thus, the possible values for x are 4 and 2.4, which satisfy the conditions for the triangle to be isosceles.
In conclusion, to have an isosceles triangle with side lengths of 3x, 5x, and 12, the values of x can be either 4 or 2.4.