Calculate Deltoid Diagonal BD: Given Area 44 cm² and AC = 11 cm

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

A deltoid is a special quadrilateral (4-sided shape) that looks like a kite! It has two pairs of adjacent sides that are equal, and its diagonals are perpendicular to each other.

Q: Why do we use the diagonal formula instead of base times height?

The diagonal formula works because deltoid diagonals are perpendicular and divide the shape into triangles. Each triangle has area = ½ × base × height, so total area = ½ × d₁ × d₂.

Q: How do I know which diagonal is which?

It doesn't matter! The formula $ Area = \frac{1}{2} \times d_1 \times d_2 $ works the same way whether AC is d₁ or d₂. Just be consistent with your labeling.

Q: What if I get a decimal answer instead of a whole number?

That's perfectly normal! Many deltoid problems have decimal answers. Just make sure to show your work clearly and round only at the very end if needed.

Q: Can I use this same formula for other quadrilaterals?

Only for special ones! This formula works for deltoids, rhombuses, and squares because their diagonals are perpendicular. For rectangles or parallelograms, use length × width instead.

Deltoid Area Formula with Known Diagonal

Given the deltoid ABCD

Side length AC equals 11 cm

The area of the deltoid is equal to 44 cm².

Find the length of the side BD

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find line BD.

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

00:18 It is diagonal one times diagonal two, divided by two.

00:23 Next, we'll plug in the given values, to solve for BD.

00:31 We'll isolate BD, by multiplying by the reciprocal. Almost there!

00:50 And that's how we find BD! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the deltoid ABCD

Side length AC equals 11 cm

The area of the deltoid is equal to 44 cm².

Find the length of the side BD

Step-by-step solution

To solve this problem, we'll use the formula for the area of a deltoid:

$\text{Area} = \frac{1}{2} \times d_1 \times d_2$

Let's work through the steps:

Step 1: Write down the formula for the area of the deltoid. The area $S$ is given as:

$44 = \frac{1}{2} \times 11 \times BD$

Step 2: Rearrange this equation to solve for the unknown diagonal $BD$ :

$44 \times 2 = 11 \times BD$

$88 = 11 \times BD$

Step 3: Divide both sides by 11 to find the length of $BD$ :

$BD = \frac{88}{11} = 8$ cm

Therefore, the solution to the problem is $BD = 8$ cm.

Final Answer

$8$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonal lengths
Technique: Substitute known values: $44 = \frac{1}{2} \times 11 \times BD$
Check: Verify by calculating: $\frac{1}{2} \times 11 \times 8 = 44$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for deltoids
Don't use base × height formula from triangles = completely wrong answer! Deltoids need the diagonal formula because they're quadrilaterals, not triangles. Always use Area = ½ × d₁ × d₂ 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 shapes?

A deltoid is a special quadrilateral (4-sided shape) that looks like a kite! It has two pairs of adjacent sides that are equal, and its diagonals are perpendicular to each other.

Why do we use the diagonal formula instead of base times height?

The diagonal formula works because deltoid diagonals are perpendicular and divide the shape into triangles. Each triangle has area = ½ × base × height, so total area = ½ × d₁ × d₂.

How do I know which diagonal is which?

It doesn't matter! The formula $Area = \frac{1}{2} \times d_1 \times d_2$ works the same way whether AC is d₁ or d₂. Just be consistent with your labeling.

What if I get a decimal answer instead of a whole number?

That's perfectly normal! Many deltoid problems have decimal answers. Just make sure to show your work clearly and round only at the very end if needed.

Can I use this same formula for other quadrilaterals?

Only for special ones! This formula works for deltoids, rhombuses, and squares because their diagonals are perpendicular. For rectangles or parallelograms, use length × width instead.

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