Calculate Diagonal Length X in a Deltoid with Area 20 cm²

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

A deltoid (also called a kite) is a special quadrilateral with two pairs of adjacent equal sides. Unlike rectangles or parallelograms, its diagonals are perpendicular and only one diagonal bisects the other.

Q: Why does the deltoid area formula use ½ × d₁ × d₂?

Because the diagonals of a deltoid are perpendicular! This creates four right triangles inside. The total area equals half the rectangle formed by the diagonals, just like with any rhombus or kite.

Q: What if I accidentally switch which diagonal is which?

No problem! Since we're multiplying the diagonals, it doesn't matter which one you call d₁ or d₂. The formula $ \frac{1}{2} \times 5 \times X $ gives the same result as $ \frac{1}{2} \times X \times 5 $.

Q: How do I solve for X when it's inside a fraction?

Work backwards from the area! Start with 20 = ½ × X × 5, then multiply both sides by 2 to get 40 = 5X, and finally divide by 5 to find X = 8.

Q: Can I check my answer without doing the whole calculation again?

Yes! Just substitute X = 8 back: $ \frac{1}{2} \times 8 \times 5 = \frac{40}{2} = 20 $ ✓ This matches the given area, so your answer is correct!

Deltoid Area Formula with Diagonal Calculations

Shown below is the deltoid ABCD.

The diagonal AC = X

Diagonal DB = 5

The area of the deltoid is 20 cm².

Calculate X.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate X

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

00:06 (diagonal times diagonal) divided by 2

00:10 We'll substitute appropriate values and solve for X

00:19 We'll multiply by 2 to eliminate the fraction

00:24 Isolate X

00:31 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

Shown below is the deltoid ABCD.

The diagonal AC = X

Diagonal DB = 5

The area of the deltoid is 20 cm².

Calculate X.

Step-by-step solution

To calculate $X$ , follow these steps:

Step 1: Use the area formula for a deltoid $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ .
Step 2: Given the area $= 20 \, \text{cm}^2$ , $d_1 = X$ , and $d_2 = 5$ , substitute these into the formula.
Step 3: Set the equation: $20 = \frac{1}{2} \times X \times 5$ .
Step 4: Simplify and solve for $X$ .

Now, let's solve:

Start with the equation $20 = \frac{1}{2} \times X \times 5$ .

This simplifies to $20 = \frac{5X}{2}$ .

Multiply both sides by 2 to eliminate the fraction:

$40 = 5X$ .

Divide both sides by 5:

$X = \frac{40}{5}$ .

Simplifying gives us $X = 8$ .

Therefore, the length of diagonal $AC$ is $X = 8 \, \text{cm}$ .

Final Answer

x=8

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Use $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ with known values
Check: Verify $\frac{1}{2} \times 8 \times 5 = 20$ matches given area ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for deltoids
Don't use triangle or rectangle formulas like base × height = wrong calculation! Deltoids are special quadrilaterals where diagonals intersect at right angles. 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 (also called a kite) is a special quadrilateral with two pairs of adjacent equal sides. Unlike rectangles or parallelograms, its diagonals are perpendicular and only one diagonal bisects the other.

Why does the deltoid area formula use ½ × d₁ × d₂?

Because the diagonals of a deltoid are perpendicular! This creates four right triangles inside. The total area equals half the rectangle formed by the diagonals, just like with any rhombus or kite.

What if I accidentally switch which diagonal is which?

No problem! Since we're multiplying the diagonals, it doesn't matter which one you call d₁ or d₂. The formula $\frac{1}{2} \times 5 \times X$ gives the same result as $\frac{1}{2} \times X \times 5$ .

How do I solve for X when it's inside a fraction?

Work backwards from the area! Start with 20 = ½ × X × 5, then multiply both sides by 2 to get 40 = 5X, and finally divide by 5 to find X = 8.

Can I check my answer without doing the whole calculation again?

Yes! Just substitute X = 8 back: $\frac{1}{2} \times 8 \times 5 = \frac{40}{2} = 20$ ✓ This matches the given area, so your answer is correct!

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Calculate Diagonal Length X in a Deltoid with Area 20 cm²

Deltoid Area Formula 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 does the deltoid area formula use ½ × d₁ × d₂?

What if I accidentally switch which diagonal is which?

How do I solve for X when it's inside a fraction?

Can I check my answer without doing the whole calculation again?

🌟 Unlock Your Math Potential

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