Quadrilateral ZYXW is a rhombus. If YX=2a+1𝑌𝑋=2𝑎+1 and if  XW=4a –25𝑋𝑊=4𝑎 –25 , find the value of a.

1. Break Down the Problem

We know that in a rhombus:

• All sides are of equal length.
• Therefore, $YX = XW$.

Given the lengths:

• $YX = 2a + 1$
• $XW = 4a - 25$

We can set the two expressions for the sides equal to each other: $2a + 1 = 4a - 25$

2. Relevant Concepts

To solve for $a$, we will isolate it in the equation. This involves moving all terms involving $a$ to one side and constant terms to the other side.

3. Analysis and Detail

Start by rearranging the equation: $2a + 1 = 4a - 25$ Subtract $2a$ from both sides: $1 = 2a - 25$ Now, add 25 to both sides: $26 = 2a$ Finally, divide both sides by 2: $a = 13$

4. Verify and Summarize

Substituting $a = 13$ back into the expressions for $YX$ and $XW$:

• For $YX$: $YX = 2(13) + 1 = 26 + 1 = 27$
• For $XW$: $XW = 4(13) - 25 = 52 - 25 = 27$ Both sides are equal, confirming our solution is correct.

Final Answer

The value of $a$ is $13$.