ABCD is a trapezium in which AD||BC, AB = 9 cm BC = 12 cm CD = 15 cm DA = 30 cm find the sum of diagonals's square

Answer

To find the sum of the squares of the diagonals in trapezium ABCD where AD || BC, we can use the formula for the sum of the squares of the diagonals in a trapezium:

\[ AC^2 + BD^2 = AB^2 + CD^2 + AD \cdot BC \]

Given:
- \( AB = 9 \, \text{cm} \)
- \( BC = 12 \, \text{cm} \)
- \( CD = 15 \, \text{cm} \)
- \( DA = 30 \, \text{cm} \)

### Step 1: Calculate \( AB^2 \) and \( CD^2 \)

\[ AB^2 = 9^2 = 81 \]
\[ CD^2 = 15^2 = 225 \]

### Step 2: Calculate \( AD \cdot BC \)

\[ AD \cdot BC = 30 \cdot 12 = 360 \]

### Step 3: Substitute values into the formula

Now, we can substitute these values into the formula:

\[ AC^2 + BD^2 = AB^2 + CD^2 + AD \cdot BC \]
\[ AC^2 + BD^2 = 81 + 225 + 360 \]

### Step 4: Add the values

\[ AC^2 + BD^2 = 81 + 225 + 360 = 666 \]

### Final Result

The sum of the squares of the diagonals \( AC^2 + BD^2 \) is \( 666 \, \text{cm}^2 \).

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