Deltoid Area Problem: Find Parameter 'a' Given Area 252 cm² and Side 7 cm

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

A deltoid (kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike a rectangle or square, its diagonals are perpendicular but not necessarily equal in length.

Q: Why do we multiply by 1/2 in the area formula?

The deltoid area formula $ \frac{1}{2} \times d_1 \times d_2 $ comes from dividing the deltoid into four right triangles. The product of diagonals gives twice the actual area, so we divide by 2.

Q: How do I identify which diagonal is which in the diagram?

Look for the labeled measurements. In this problem, AC = 7 cm is one diagonal, and BD = 9a is the other diagonal. The diagonals of a deltoid always intersect at right angles.

Q: What if I get a decimal answer for the parameter?

Check your arithmetic carefully! In this problem, $ a = \frac{252}{31.5} = 8 $ gives a whole number. If you get a decimal, verify your calculation and make sure you used the correct formula.

Deltoid Area with Parameter Calculation

Given the deltoid ABCD

Side length AC equals 7 cm

The area of the deltoid is equal to 252 cm².

Find the value of the parameter $a$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find A

00:03 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 A

00:22 Isolate A

00:42 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

Given the deltoid ABCD

Side length AC equals 7 cm

The area of the deltoid is equal to 252 cm².

Find the value of the parameter $a$

Step-by-step solution

To solve this problem, we will calculate the value of $a$ using the given area of the deltoid and the known diagonal AC.

The formula for the area of a deltoid (kite) is:

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

Here, $d_1 = AC = 7$ cm, and $d_2 = BD = 9a$ cm. The area $S = 252$ cm².

Substitute the known values into the area formula:

$252 = \frac{1}{2} \times 7 \times 9a$

Simplify and solve for $a$ :

$252 = \frac{63}{2} \times a$

$252 = 31.5a$

$a = \frac{252}{31.5}$

$a = 8$

Therefore, the value of the parameter $a$ is $8$ .

Final Answer

$8$

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonals
Technique: Substitute $252 = \frac{1}{2} \times 7 \times 9a$ and solve
Check: Verify $\frac{1}{2} \times 7 \times 72 = 252$ cm² ✓

Common Mistakes

Avoid these frequent errors

Forgetting to use the factor 1/2 in deltoid area formula
Don't calculate $7 \times 9a = 252$ directly = wrong answer of a = 4! This ignores that deltoid area is half the diagonal product. Always use $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ .

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 exactly is a deltoid and how is it different from other quadrilaterals?

A deltoid (kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike a rectangle or square, its diagonals are perpendicular but not necessarily equal in length.

Why do we multiply by 1/2 in the area formula?

The deltoid area formula $\frac{1}{2} \times d_1 \times d_2$ comes from dividing the deltoid into four right triangles. The product of diagonals gives twice the actual area, so we divide by 2.

How do I identify which diagonal is which in the diagram?

Look for the labeled measurements. In this problem, AC = 7 cm is one diagonal, and BD = 9a is the other diagonal. The diagonals of a deltoid always intersect at right angles.

What if I get a decimal answer for the parameter?

Check your arithmetic carefully! In this problem, $a = \frac{252}{31.5} = 8$ gives a whole number. If you get a decimal, verify your calculation and make sure you used the correct formula.

Can I solve this problem without knowing the area formula?

No - you need the deltoid area formula! Unlike triangles where you might use base × height, deltoids require the specific formula using both diagonal lengths.

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