Deltoid Area Problem: Finding AC Length When Area is 27 cm²

To solve this problem, we'll use the formula relating the area of a deltoid to its diagonals:

• Step 1: Identify the given details:
• Diagonal AC = $X$.
• Diagonal DB = $3X$.
• Area = 27 cm$^2$.
• Step 2: Write down the formula for the area:

The area of a deltoid is given by:

$\text{Area} = \frac{1}{2} \times AC \times DB$

• Step 3: Substitute the values into the formula:

Substitute $AC = X$ and $DB = 3X$ into the equation:

$\frac{1}{2} \times X \times 3X = 27$

• Step 4: Simplify and solve for $X$:

First, simplify the left side:

$\frac{1}{2} \times 3X^2 = \frac{3}{2}X^2$

Thus, the equation becomes:

$\frac{3}{2}X^2 = 27$

Multiply both sides by 2 to clear the fraction:

$3X^2 = 54$

Divide both sides by 3:

$X^2 = 18$

Take the square root of both sides:

$X = \sqrt{18}$

Therefore, the length of diagonal AC is $\sqrt{18}$.