Find the Side Length BD in a Deltoid with Area 48 cm² and AC = 6 cm

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

A deltoid is a special quadrilateral (4-sided shape) where the diagonals are perpendicular to each other. Unlike rectangles or parallelograms, deltoids use the diagonal formula for area calculations.

Q: Why do we use ½ × d₁ × d₂ instead of base × height?

The diagonal formula works because perpendicular diagonals create four right triangles inside the deltoid. Each triangle has area = ½ × base × height, so total area = ½ × diagonal₁ × diagonal₂.

Q: How do I know which diagonal is which in the problem?

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 d₁ or d₂ since multiplication is commutative.

Q: What if I get stuck on the algebra after substituting?

Break it down step by step: multiply both sides by 2 first to eliminate the fraction, then divide by the known diagonal length. Always do the same operation to both sides!

Q: Can I use this formula for any quadrilateral?

No! This formula only works when the diagonals are perpendicular (meet at 90°). For other quadrilaterals like rectangles, you need different formulas.

Deltoid Area Formula with Diagonal Calculation

The deltoid ABCD is shown below.

Side length AC equals 6 cm.

The area of the deltoid is 48 cm².

What is the length of the side BD?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find BD

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

00:07 (diagonal times diagonal) divided by 2

00:12 We'll substitute appropriate values according to the given data and solve for BD

00:20 Divide 6 by 2

00:25 Isolate BD

00:35 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.

Side length AC equals 6 cm.

The area of the deltoid is 48 cm².

What is the length of the side BD?

Step-by-step solution

To solve for $BD$ , the diagonal of the deltoid, follow these steps:

Step 1: Recognize that the area of a deltoid with perpendicular diagonals is given by the formula $A = \frac{1}{2} \times d_1 \times d_2$ .
Step 2: Substitute the known values into the formula: $48 = \frac{1}{2} \times 6 \times BD$ .
Step 3: Simplify and solve for $BD$ :

Substituting $AC = 6$ cm, we have:

$48 = \frac{1}{2} \times 6 \times BD$

Multiply both sides by 2 to clear the fraction:

$96 = 6 \times BD$

Divide both sides by 6 to solve for $BD$ :

$BD = \frac{96}{6} = 16$ cm

Thus, the length of $BD$ is $16$ cm.

Final Answer

$16$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Substitute known values: $48 = \frac{1}{2} \times 6 \times BD$
Check: Verify: $\frac{1}{2} \times 6 \times 16 = 48$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using incorrect area formula for deltoid
Don't use triangle area formula base × height ÷ 2 = wrong calculation! Deltoids need the diagonal formula because they're quadrilaterals with perpendicular diagonals. Always use Area = ½ × diagonal₁ × diagonal₂ for deltoids.

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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 special quadrilateral (4-sided shape) where the diagonals are perpendicular to each other. Unlike rectangles or parallelograms, deltoids use the diagonal formula for area calculations.

Why do we use ½ × d₁ × d₂ instead of base × height?

The diagonal formula works because perpendicular diagonals create four right triangles inside the deltoid. Each triangle has area = ½ × base × height, so total area = ½ × diagonal₁ × diagonal₂.

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

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 d₁ or d₂ since multiplication is commutative.

What if I get stuck on the algebra after substituting?

Break it down step by step: multiply both sides by 2 first to eliminate the fraction, then divide by the known diagonal length. Always do the same operation to both sides!

Can I use this formula for any quadrilateral?

No! This formula only works when the diagonals are perpendicular (meet at 90°). For other quadrilaterals like rectangles, you need different formulas.

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