In parallelogram ABCD , AB = (3x- 4) cm, BC = ( y- 1) cm , CD = ( y+ 5) cm and AD = (2x+ 5) cm. find the ratio AB : BC.
Find the values of x and y.

Answer

To solve the problem step by step, we will use the properties of a parallelogram, where opposite sides are equal.

### Step 1: Set up the equations based on the properties of the parallelogram.

In parallelogram ABCD:
- \( AB = CD \)
- \( AD = BC \)

Given:
- \( AB = 3x - 4 \) cm
- \( BC = y - 1 \) cm
- \( CD = y + 5 \) cm
- \( AD = 2x + 5 \) cm

From the properties:
1. \( AB = CD \) gives us the equation:
\[ 3x - 4 = y + 5 \]
2. \( AD = BC \) gives us the equation:
\[ 2x + 5 = y - 1 \]

### Step 2: Simplify the equations.

**From Equation 1:**
\[ 3x - 4 = y + 5 \]
Rearranging gives:
\[ 3x - y - 9 = 0 \quad \text{(Equation 3)} \]

**From Equation 2:**
\[ 2x + 5 = y - 1 \]
Rearranging gives:
\[ 2x - y + 6 = 0 \quad \text{(Equation 4)} \]

### Step 3: Solve the system of equations.

We have two equations:
1. \( 3x - y - 9 = 0 \) (Equation 3)
2. \( 2x - y + 6 = 0 \) (Equation 4)

To eliminate \( y \), we can subtract Equation 4 from Equation 3:
\[ (3x - y - 9) - (2x - y + 6) = 0 \]
This simplifies to:
\[ 3x - y - 9 - 2x + y - 6 = 0 \]
Combining like terms gives:
\[ x - 15 = 0 \]
Thus, we find:
\[ x = 15 \]

### Step 4: Substitute \( x \) back into one of the equations to find \( y \).

Using Equation 3:
\[ 3(15) - y - 9 = 0 \]
Calculating gives:
\[ 45 - y - 9 = 0 \]
This simplifies to:
\[ 36 - y = 0 \quad \Rightarrow \quad y = 36 \]

### Step 5: Find the ratio \( AB : BC \).

Now we can find the ratio:
\[ AB = 3x - 4 = 3(15) - 4 = 45 - 4 = 41 \text{ cm} \]
\[ BC = y - 1 = 36 - 1 = 35 \text{ cm} \]
Thus, the ratio \( AB : BC \) is:
\[ \frac{AB}{BC} = \frac{41}{35} \]

### Final Answers:
- Values: \( x = 15 \), \( y = 36 \)
- Ratio \( AB : BC = 41 : 35 \)

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