Deltoid Problem: Solve for 'a' Given Area 6a and Diagonals 2a+2 and a

Q: What's the difference between a deltoid and a kite?

A deltoid and a kite are actually the same shape! Both have two pairs of adjacent sides that are equal, and the same area formula applies: $ \frac{d_1 \times d_2}{2} $.

Q: Why do I get a quadratic equation when solving this?

When you expand $ a(2a+2) = 12a $, you get $ 2a^2 + 2a = 12a $. This creates a quadratic because you're multiplying 'a' by an expression containing 'a'. Fortunately, you can divide by 'a' since a ≠ 0!

Q: Can I solve this without expanding the parentheses?

Yes! After getting $ a(2a+2) = 12a $, you can divide both sides by 'a' first: $ 2a+2 = 12 $. Then solve: $ 2a = 10 $, so $ a = 5 $.

Q: What if my answer gives negative area or diagonal lengths?

Since we're dealing with geometric measurements, both diagonals and area must be positive. If you get a negative value for 'a', check your algebra - you might have made a sign error!

Deltoid Area with Diagonal Expressions

Given the deltoid ABCD

The main diagonal is equal to 2a+2

Secondary diagonal is equal to a

The area of the deltoid equals 6a

Calculate a a

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find A

00:03 Use the formula for calculating the area of a kite

00:07 Product of diagonals divided by 2

00:17 Substitute appropriate values according to the given data, and solve for A

00:37 Multiply by 2 to eliminate the fraction

00:41 Open parentheses properly, multiply by each factor

00:50 Arrange the equation so that one side equals 0

01:06 Factor out common terms from the parentheses

01:12 Find the possible solutions

01:15 A must be positive since it represents a physical length of a side

01:20 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the deltoid ABCD

The main diagonal is equal to 2a+2

Secondary diagonal is equal to a

The area of the deltoid equals 6a

Calculate a a

Step-by-step solution

To solve the question, we first need to remember the formula for the area of a kite:

Diagonal * Diagonal / 2

This means that if we substitute the given data we can see that:

a(2a+2)/2 = area of the kite

Let's remember that we are also given the area, so we'll put that in the equation too

a(2a+2)/2 = 6a

Now we have an equation that we can easily solve.

First, let's get rid of the fraction, so we'll multiply both sides of the equation by 2

a(2a+2)=6a*2
a(2a+2)=12a

Let's expand the parentheses on the left side of the equation

2a²+2a=12a

2a²=10a

Let's divide both sides of the equation by a

2a=10

Let's divide again by 2

a=5

And that's the solution!

Final Answer

5 cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals diagonal₁ × diagonal₂ ÷ 2
Technique: Substitute $\frac{a(2a+2)}{2} = 6a$ then expand and simplify
Check: Verify a=5: diagonals 12 and 5 give area 30 = 6(5) ✓

Common Mistakes

Avoid these frequent errors

Forgetting to divide by 2 in area formula
Don't use diagonal₁ × diagonal₂ = area formula directly = double the correct area! This gives wrong equations like a(2a+2) = 6a instead of a(2a+2)/2 = 6a. Always remember the deltoid area formula includes division by 2.

Practice Quiz

Test your knowledge with interactive questions

Look at the kite ABCD below.

Diagonal DB = 10

CB = 4

Is it possible to calculate the area of the kite? If so, what is it?

FAQ

Everything you need to know about this question

What's the difference between a deltoid and a kite?

A deltoid and a kite are actually the same shape! Both have two pairs of adjacent sides that are equal, and the same area formula applies: $\frac{d_1 \times d_2}{2}$ .

Why do I get a quadratic equation when solving this?

When you expand $a(2a+2) = 12a$ , you get $2a^2 + 2a = 12a$ . This creates a quadratic because you're multiplying 'a' by an expression containing 'a'. Fortunately, you can divide by 'a' since a ≠ 0!

Can I solve this without expanding the parentheses?

Yes! After getting $a(2a+2) = 12a$ , you can divide both sides by 'a' first: $2a+2 = 12$ . Then solve: $2a = 10$ , so $a = 5$ .

What if my answer gives negative area or diagonal lengths?

Since we're dealing with geometric measurements, both diagonals and area must be positive. If you get a negative value for 'a', check your algebra - you might have made a sign error!

How do I verify my answer is correct?

Substitute a = 5 back into the original information: Main diagonal = 2(5) + 2 = 12, Secondary diagonal = 5, Area = $\frac{12 \times 5}{2} = 30$ . Check: 6a = 6(5) = 30 ✓

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Deltoid Problem: Solve for 'a' Given Area 6a and Diagonals 2a+2 and a

Deltoid Area with Diagonal Expressions

❤️ 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's the difference between a deltoid and a kite?

Why do I get a quadratic equation when solving this?

Can I solve this without expanding the parentheses?

What if my answer gives negative area or diagonal lengths?

How do I verify my answer is correct?

🌟 Unlock Your Math Potential

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