G 3x-10 5x-8 F 4x+1 H
The Correct Answer and Explanation is:
The correct answer is that the problem presented in the image has no valid solution. The information given is mathematically inconsistent, making it impossible to solve for a value of x that satisfies the geometric conditions.
The image shows a triangle labeled FGH. On each of the three sides, there is a small tick mark. In geometry, these identical markings are used to indicate that the sides are of equal length. A triangle with three equal sides is known as an equilateral triangle. For this triangle to be equilateral as depicted, the algebraic expressions representing the lengths of its three sides must be equal to one another.
The lengths of the sides are given by the following expressions:
Side FG = 3x – 10
Side GH = 5x – 8
Side FH = 4x + 1
To find a value for x that makes the triangle equilateral, we must be able to set the expressions for any two sides equal to each other and get the same value for x in all cases. Let’s examine the three possible equations:
- If we set side FG equal to side GH, the equation is 3x – 10 = 5x – 8. To solve this, we can subtract 3x from both sides, which gives -10 = 2x – 8. Adding 8 to both sides results in -2 = 2x. Dividing by 2, we find that x = -1.
- If we set side GH equal to side FH, the equation is 5x – 8 = 4x + 1. To solve this, we subtract 4x from both sides, which gives x – 8 = 1. Adding 8 to both sides, we find that x = 9.
- If we set side FG equal to side FH, the equation is 3x – 10 = 4x + 1. Subtracting 3x from both sides gives -10 = x + 1. Subtracting 1 from both sides, we find that x = -11.
We have found three different values for x ( -1, 9, and -11). Because there is no single, consistent value for x that makes all three sides equal, the problem is flawed. Furthermore, if we were to use the values x = -1 or x = -11, some of the calculated side lengths would be negative, which is impossible for a physical triangle. For example, if x = -1, the length of side FH would be 4(-1) + 1 = -3. Therefore, the problem contains a fundamental contradiction and cannot be solved as stated.thumb_upthumb_down
