Calculate Deltoid Dimensions: Finding DB When AC = 2X and Area = 32 cm²

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

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Its diagonals are perpendicular, which is why we can use the formula Area = (1/2) × d₁ × d₂.

Q: Why don't we simplify √32 to a decimal?

The exact answer $ \sqrt{32} $ is more precise than a decimal approximation. In mathematics, we often keep answers in radical form unless specifically asked to round.

Q: Can I solve this by setting up the equation differently?

Yes! You could write $ 32 = \frac{1}{2} \times AC \times DB $ first, then substitute. The key is always using the correct area formula for deltoids.

Q: What if I got X = ±√32? Should I include the negative?

Since we're dealing with a length measurement, we only take the positive square root. Distances cannot be negative, so DB = $ \sqrt{32} $ cm.

Area Formulas with Variable Diagonals

Shown below is the deltoid ABCD.

AC = 2X

DB = X

The area of the deltoid is equal to 32 cm².

Calculate DB.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's start by calculating the diagonal D to B.

00:15 We'll use the formula for finding a rhombus's area.

00:20 That's diagonal one times diagonal two, divided by two.

00:25 Let's substitute the given values to find diagonal D to B.

00:33 Then, multiply by two to remove the fraction.

00:43 Next, we'll isolate the variable X.

00:56 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Shown below is the deltoid ABCD.

AC = 2X

DB = X

The area of the deltoid is equal to 32 cm².

Calculate DB.

Step-by-step solution

To solve this problem, we will utilize the formula for the area of a deltoid, which is $\frac{1}{2} \times AC \times DB$ . Given:

Diagonal $AC = 2X$
Diagonal $DB = X$
Area = $32 \, \text{cm}^2$

The formula for the area of the deltoid is:

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

Substitute the given values into the formula:

$32 = \frac{1}{2} \times 2X \times X$

Simplify the equation:

$32 = \frac{1}{2} \times 2X^2$

$32 = X^2$

Solve for $X$ by taking the square root of both sides:

$X = \sqrt{32}$

Since $DB = X$ , the length of diagonal $DB$ is $\sqrt{32}$ .

Thus, the solution to the problem is $\sqrt{32}$ .

Final Answer

$\sqrt{32}$

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Substitute AC = 2X and DB = X into $\frac{1}{2} \times 2X \times X$
Check: Verify $\sqrt{32} \times 2\sqrt{32} \times \frac{1}{2} = 32$ ✓

Common Mistakes

Avoid these frequent errors

Using perimeter formula instead of area formula
Don't add the diagonal lengths AC + DB = 32! This confuses perimeter with area and gives 2X + X = 32, leading to X = 32/3. Always use Area = (1/2) × diagonal₁ × diagonal₂ for deltoids.

Practice Quiz

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FAQ

Everything you need to know about this question

What exactly is a deltoid and how is it different from other quadrilaterals?

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Its diagonals are perpendicular, which is why we can use the formula Area = (1/2) × d₁ × d₂.

Why don't we simplify √32 to a decimal?

The exact answer $\sqrt{32}$ is more precise than a decimal approximation. In mathematics, we often keep answers in radical form unless specifically asked to round.

Can I solve this by setting up the equation differently?

Yes! You could write $32 = \frac{1}{2} \times AC \times DB$ first, then substitute. The key is always using the correct area formula for deltoids.

How do I know which diagonal is which in the diagram?

Look at the labels carefully! AC connects vertices A and C (vertical diagonal = 2X), while DB connects vertices D and B (horizontal diagonal = X).

What if I got X = ±√32? Should I include the negative?

Since we're dealing with a length measurement, we only take the positive square root. Distances cannot be negative, so DB = $\sqrt{32}$ cm.

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Calculate Deltoid Dimensions: Finding DB When AC = 2X and Area = 32 cm²

Area Formulas with Variable Diagonals

❤️ 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 exactly is a deltoid and how is it different from other quadrilaterals?

Why don't we simplify √32 to a decimal?

Can I solve this by setting up the equation differently?

How do I know which diagonal is which in the diagram?

What if I got X = ±√32? Should I include the negative?

🌟 Unlock Your Math Potential

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