Calculate AC Length in Deltoid ABCD with Area 28 cm² and Diagonal DB = 4

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

A deltoid is a kite-shaped quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals, which is why we use the special diagonal formula for area.

Q: Why do we use ½ × diagonal₁ × diagonal₂ for deltoid area?

This formula works because the diagonals of a deltoid are perpendicular (meet at 90°). They divide the deltoid into four right triangles, and this formula calculates the total area of all four triangles together.

Q: How do I remember which diagonal is which?

It doesn't matter! The area formula $ A = \frac{1}{2} \times d_1 \times d_2 $ works the same way regardless of which diagonal you call AC or DB. Multiplication is commutative.

Q: What if I'm given the area and need to find a diagonal?

Rearrange the formula! If $ A = \frac{1}{2} \times AC \times DB $, then $ AC = \frac{2A}{DB} $. Just like in this problem: $ AC = \frac{2 \times 28}{4} = 14 $ cm.

Q: Can I use this formula for any quadrilateral?

No! This formula only works for deltoids (kites) and rhombuses because they have perpendicular diagonals. For other quadrilaterals, you need different area formulas.

Deltoid Area Formulas with Diagonal Calculations

Shown below is the deltoid ABCD.

DB = 4

The area of the deltoid is 28 cm².

Calculate the length of side AC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's find the length of A C.

00:14 We'll use the formula for the area of a kite.

00:17 It's diagonal times diagonal. Then, divide by 2.

00:22 Now, let's put in the values we have, and find A C.

00:34 Next, multiply by 2 to get rid of the fraction.

00:42 Now, let's isolate A C to find its length.

00:51 And that's how we solve this problem!

Step-by-step written solution

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Understand the problem

Shown below is the deltoid ABCD.

DB = 4

The area of the deltoid is 28 cm².

Calculate the length of side AC.

Step-by-step solution

To calculate the length of the diagonal $AC$ , we start by using the area formula for a deltoid, which involves its diagonals. The area $A$ of a deltoid is given by:

A = \frac{1}{2} \times AC \times DB

Given:

The area of the deltoid $A = 28 \, \text{cm}^2$ .
The length of diagonal $DB = 4 \, \text{cm}$ .

We can plug these values into the formula:

28 = \frac{1}{2} \times AC \times 4

Solving for $AC$ :

28 = 2 \times AC

Divide both sides by $2$ :

AC = \frac{28}{2} = 14

Therefore, the length of side $AC$ is 14 cm.

Final Answer

14 cm²

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonals
Technique: From $28 = \frac{1}{2} \times AC \times 4$ , solve for AC
Check: Verify $\frac{1}{2} \times 14 \times 4 = 28$ cm² ✓

Common Mistakes

Avoid these frequent errors

Confusing deltoid area formula with rectangle or triangle formulas
Don't use Area = length × width or Area = ½ × base × height for deltoids = wrong answer! These formulas don't apply to deltoids. Always use the deltoid-specific formula: Area = ½ × diagonal₁ × diagonal₂.

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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 shapes?

A deltoid is a kite-shaped quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or parallelograms, deltoids have perpendicular diagonals, which is why we use the special diagonal formula for area.

Why do we use ½ × diagonal₁ × diagonal₂ for deltoid area?

This formula works because the diagonals of a deltoid are perpendicular (meet at 90°). They divide the deltoid into four right triangles, and this formula calculates the total area of all four triangles together.

How do I remember which diagonal is which?

It doesn't matter! The area formula $A = \frac{1}{2} \times d_1 \times d_2$ works the same way regardless of which diagonal you call AC or DB. Multiplication is commutative.

What if I'm given the area and need to find a diagonal?

Rearrange the formula! If $A = \frac{1}{2} \times AC \times DB$ , then $AC = \frac{2A}{DB}$ . Just like in this problem: $AC = \frac{2 \times 28}{4} = 14$ cm.

Can I use this formula for any quadrilateral?

No! This formula only works for deltoids (kites) and rhombuses because they have perpendicular diagonals. For other quadrilaterals, you need different area formulas.

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Calculate AC Length in Deltoid ABCD with Area 28 cm² and Diagonal DB = 4

Deltoid Area Formulas with Diagonal Calculations

❤️ 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 shapes?

Why do we use ½ × diagonal₁ × diagonal₂ for deltoid area?

How do I remember which diagonal is which?

What if I'm given the area and need to find a diagonal?

Can I use this formula for any quadrilateral?

🌟 Unlock Your Math Potential

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