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

Q: What is a deltoid and how is it different from other shapes?

A deltoid is a quadrilateral (4-sided shape) with two pairs of adjacent sides that are equal. It's also called a kite! Its diagonals are perpendicular, which is why we can use the area formula $ \frac{1}{2} \times d_1 \times d_2 $.

Q: Why do we get √18 as the answer instead of a whole number?

Square roots are exact answers! $ \sqrt{18} $ is more precise than the decimal 4.24... You could simplify it to $ 3\sqrt{2} $, but both forms are correct.

Q: Can I use this formula for any quadrilateral?

No! This formula only works when the diagonals are perpendicular (meet at 90°). This applies to deltoids, squares, rhombuses, and kites, but not rectangles or parallelograms.

Q: How do I know which diagonal is which?

It doesn't matter! The area formula $ \frac{1}{2} \times d_1 \times d_2 $ works regardless of which diagonal you call $ d_1 $ or $ d_2 $ since multiplication is commutative.

Q: What if I can't take the square root perfectly?

That's normal! Many geometry problems have irrational answers. Leave your answer as $ \sqrt{18} $ or simplify to $ 3\sqrt{2} $ - both are exact and correct!

Deltoid Area Formula with Diagonal Variables

The deltoid ABCD is shown below.

AC = X

DB = 3X

The area of the deltoid is 27 cm².

Calculate the length of AC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate AC

00:03 We'll use the formula for calculating the area of a kite

00:07 (diagonal times diagonal) divided by 2

00:14 Let's substitute appropriate values according to the given data and find AC

00:17 Multiply by 2 to eliminate the fraction

00:25 Isolate X

00:39 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The deltoid ABCD is shown below.

AC = X

DB = 3X

The area of the deltoid is 27 cm².

Calculate the length of AC.

Step-by-step solution

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 $\sqrt{18}$ .

Final Answer

$\sqrt{18}$

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonals
Technique: Substitute AC = X and DB = 3X into $\frac{1}{2} \times X \times 3X = 27$
Check: $\frac{1}{2} \times \sqrt{18} \times 3\sqrt{18} = \frac{1}{2} \times 54 = 27$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply diagonals by 1/2
Don't use Area = AC × DB directly = 54 instead of 27! This gives double the actual area because you missed the 1/2 factor. Always use the complete formula: Area = $\frac{1}{2} \times d_1 \times d_2$ for any quadrilateral with perpendicular diagonals.

Practice Quiz

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Indicate the correct answer

The next quadrilateral is:

FAQ

Everything you need to know about this question

What is a deltoid and how is it different from other shapes?

A deltoid is a quadrilateral (4-sided shape) with two pairs of adjacent sides that are equal. It's also called a kite! Its diagonals are perpendicular, which is why we can use the area formula $\frac{1}{2} \times d_1 \times d_2$ .

Why do we get √18 as the answer instead of a whole number?

Square roots are exact answers! $\sqrt{18}$ is more precise than the decimal 4.24... You could simplify it to $3\sqrt{2}$ , but both forms are correct.

Can I use this formula for any quadrilateral?

No! This formula only works when the diagonals are perpendicular (meet at 90°). This applies to deltoids, squares, rhombuses, and kites, but not rectangles or parallelograms.

How do I know which diagonal is which?

It doesn't matter! The area formula $\frac{1}{2} \times d_1 \times d_2$ works regardless of which diagonal you call $d_1$ or $d_2$ since multiplication is commutative.

What if I can't take the square root perfectly?

That's normal! Many geometry problems have irrational answers. Leave your answer as $\sqrt{18}$ or simplify to $3\sqrt{2}$ - both are exact and correct!

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

Deltoid Area Formula with Diagonal Variables

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What is a deltoid and how is it different from other shapes?

Why do we get √18 as the answer instead of a whole number?

Can I use this formula for any quadrilateral?

How do I know which diagonal is which?

What if I can't take the square root perfectly?

🌟 Unlock Your Math Potential

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