What value of x will make parallelogram ABCD a rhombus?
x =
The Correct Answer and Explanation is :
To determine the value of ( x ) that will make parallelogram ABCD a rhombus, we need to use the defining property of a rhombus: all four sides of a rhombus are congruent.
Problem Analysis
From the image:
- Parallelogram ABCD has sides labeled with algebraic expressions:
- One pair of opposite sides is ( 2x + 3 ) and the other is ( 3x – 4 ).
- To make ABCD a rhombus, the side lengths must all be equal:
[
2x + 3 = 3x – 4
]
Solving the Equation
Step 1: Set the side expressions equal to each other:
[
2x + 3 = 3x – 4
]
Step 2: Subtract ( 2x ) from both sides to isolate ( x ) terms on one side:
[
3 = x – 4
]
Step 3: Add ( 4 ) to both sides to solve for ( x ):
[
x = 7
]
Verification
If ( x = 7 ):
- The length of one side is:
[
2x + 3 = 2(7) + 3 = 14 + 3 = 17
] - The length of the other side is:
[
3x – 4 = 3(7) – 4 = 21 – 4 = 17
]
Since both sides are equal, all four sides of the parallelogram will be 17, confirming it is a rhombus.
Explanation
A parallelogram becomes a rhombus when all its sides are of equal length. By setting the expressions for adjacent sides equal, we solved for ( x ) and verified that it satisfies the condition of a rhombus. This method uses basic algebra and the geometric property of a rhombus, ensuring logical and accurate results.
Thus, the correct value of ( x ) is:
[
\boxed{7}
]