Calculate Deltoid Area: Given 10cm Diagonal and 40% Length Relationship

Q: What does '40% longer' actually mean in this problem?

'40% longer' means the diagonal AC is the original length DB plus 40% more. So AC = DB + (0.4 × DB) = 10 + 4 = 14 cm, not just 4 cm!

Q: Why do we use the diagonal formula for a deltoid's area?

A deltoid (kite) has perpendicular diagonals, just like a rhombus. The area formula $ \frac{1}{2} \times d_1 \times d_2 $ works for any quadrilateral with perpendicular diagonals.

Q: How can I remember which diagonal is which?

Look at the diagram carefully! The problem states DB = 10 and that AC is 40% longer. Always identify the given diagonal first, then calculate the other one.

Q: What if I calculated AC = 10 × 1.4 = 14 cm instead?

That's actually correct too! When something is 40% longer, it's 140% of the original (100% + 40% = 140% = 1.4). Both methods give AC = 14 cm.

Q: Can I use this formula for any quadrilateral?

No! This diagonal formula only works for quadrilaterals with perpendicular diagonals, like kites, rhombuses, and squares. Regular quadrilaterals need different area formulas.

Deltoid Area with Percentage-Based Diagonal Relationships

Below is the deltoid ABCD.

Given in cm: DB = 10

Diagonal AC is 40% longer than diagonal DB.

Calculate the area of the deltoid.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's find the area of the kite.

00:15 To do this, we'll use the formula for area of a kite.

00:19 It's diagonal one times diagonal two, divided by 2.

00:24 From the given data, let's find the ratio of the diagonals.

00:35 We will substitute the value of diagonal BD, given in the data, and then solve for diagonal AC.

00:43 First, we'll change the percentages into fractions to make it easier.

00:51 This gives us the length of diagonal AC.

00:59 Now, let's use these values in the area formula and solve it step by step.

01:11 And that's how we find the area of the kite!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Below is the deltoid ABCD.

Given in cm: DB = 10

Diagonal AC is 40% longer than diagonal DB.

Calculate the area of the deltoid.

Step-by-step solution

To solve the problem, we need to find the lengths of the diagonals and then calculate the area of the deltoid using these lengths.

First, find the length of diagonal $AC$ .

Given that $AC$ is 40% longer than $DB$ , which is 10 cm:

$AC = 10 + 0.4 \times 10 = 10 + 4 = 14 \, \text{cm}$

Now, apply the formula for the area of the deltoid:

The area $A$ of a deltoid can be calculated using the formula:

$A = \frac{1}{2} \times d_1 \times d_2$

Substituting the given values ( $d_1 = DB = 10 \, \text{cm}$ and $d_2 = AC = 14 \, \text{cm}$ ):

$A = \frac{1}{2} \times 10 \times 14 = \frac{1}{2} \times 140 = 70 \, \text{cm}^2$

Therefore, the area of the deltoid ABCD is 70 cm², which matches the given correct answer.

Final Answer

70 cm²

Key Points to Remember

Essential concepts to master this topic

Formula: Area equals half the product of both diagonal lengths
Technique: Calculate AC = 10 + (40% × 10) = 14 cm
Check: Verify $\frac{1}{2} \times 10 \times 14 = 70$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using the percentage as a direct multiplier instead of calculating the increase
Don't calculate AC as 10 × 40% = 4 cm! This gives only the increase amount, not the total length. The percentage means AC is 40% MORE than DB. Always calculate: original length + (percentage × original length).

Practice Quiz

Test your knowledge with interactive questions

Look at the deltoid in the figure:

What is its area?

FAQ

Everything you need to know about this question

What does '40% longer' actually mean in this problem?

'40% longer' means the diagonal AC is the original length DB plus 40% more. So AC = DB + (0.4 × DB) = 10 + 4 = 14 cm, not just 4 cm!

Why do we use the diagonal formula for a deltoid's area?

A deltoid (kite) has perpendicular diagonals, just like a rhombus. The area formula $\frac{1}{2} \times d_1 \times d_2$ works for any quadrilateral with perpendicular diagonals.

How can I remember which diagonal is which?

Look at the diagram carefully! The problem states DB = 10 and that AC is 40% longer. Always identify the given diagonal first, then calculate the other one.

What if I calculated AC = 10 × 1.4 = 14 cm instead?

That's actually correct too! When something is 40% longer, it's 140% of the original (100% + 40% = 140% = 1.4). Both methods give AC = 14 cm.

Can I use this formula for any quadrilateral?

No! This diagonal formula only works for quadrilaterals with perpendicular diagonals, like kites, rhombuses, and squares. Regular quadrilaterals need different area formulas.

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Calculate Deltoid Area: Given 10cm Diagonal and 40% Length Relationship

Deltoid Area with Percentage-Based Diagonal Relationships

❤️ 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 does '40% longer' actually mean in this problem?

Why do we use the diagonal formula for a deltoid's area?

How can I remember which diagonal is which?

What if I calculated AC = 10 × 1.4 = 14 cm instead?

Can I use this formula for any quadrilateral?

🌟 Unlock Your Math Potential

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