Calculate Side Length AC in a Deltoid with Area 45 cm² and DB = 9

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

A deltoid (or kite) is a quadrilateral with two pairs of adjacent equal sides. Unlike rectangles or squares, its area depends on the diagonals that intersect at right angles, not base and height.

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

The diagonals of a deltoid are perpendicular, which creates four right triangles. The formula $ S = \frac{1}{2} \times d_1 \times d_2 $ calculates the total area of these triangles efficiently.

Q: How do I know which measurements are the diagonals?

Diagonals connect opposite vertices and cross inside the shape. In this problem, AC and DB are diagonals because they connect corners A-C and D-B respectively.

Deltoid Area with Diagonal Calculations

Look at the deltoid ABCD below.

DB = 9

The area of the deltoid is equal to 45 cm².

Calculate the length of side AC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate diagonal AC

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

00:07 (diagonal times diagonal) divided by 2

00:10 We'll substitute appropriate values according to the given data and find AC

00:23 We'll multiply by 2 to eliminate the fraction

00:30 We'll isolate AC

00:35 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

Look at the deltoid ABCD below.

DB = 9

The area of the deltoid is equal to 45 cm².

Calculate the length of side AC.

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Identify the given information
Step 2: Use the formula for the area of a deltoid
Step 3: Perform the algebraic solution

Let's work through each step:

Step 1: We know that $DB = 9$ cm and the area $S = 45$ cm². We are asked to find $AC$ .

Step 2: The formula for the area of a deltoid is $S = \frac{1}{2} \times d_1 \times d_2$ . Here, $d_1 = AC$ and $d_2 = DB = 9$ .

Step 3: Substitute the known values into the formula:
$45 = \frac{1}{2} \times AC \times 9$
$45 = \frac{9}{2} \times AC$
Multiply both sides by 2 to eliminate the fraction:
$90 = 9 \times AC$
Divide both sides by 9 to solve for $AC$ :
$AC = \frac{90}{9}$
$AC = 10$ cm.

Therefore, the length of side $AC$ is 10 cm.

Final Answer

10 cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonals
Technique: $45 = \frac{1}{2} \times AC \times 9$ becomes $90 = 9 \times AC$
Check: Verify $\frac{1}{2} \times 10 \times 9 = 45$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for deltoid
Don't use base × height formula from rectangles = completely wrong answer! Deltoids aren't rectangles, so this gives impossible results like negative lengths. Always use the diagonal formula: $S = \frac{1}{2} \times d_1 \times d_2$ .

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 (or kite) is a quadrilateral with two pairs of adjacent equal sides. Unlike rectangles or squares, its area depends on the diagonals that intersect at right angles, not base and height.

Why do we use the diagonal formula for deltoid area?

The diagonals of a deltoid are perpendicular, which creates four right triangles. The formula $S = \frac{1}{2} \times d_1 \times d_2$ calculates the total area of these triangles efficiently.

How do I know which measurements are the diagonals?

Diagonals connect opposite vertices and cross inside the shape. In this problem, AC and DB are diagonals because they connect corners A-C and D-B respectively.

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

Decimal answers are perfectly valid! Many real-world measurements aren't whole numbers. Just make sure to round appropriately based on the precision of your given measurements.

Can I solve this problem without using the area formula?

No - the area formula is essential because it's the only relationship connecting the given information (area = 45 cm², DB = 9 cm) to what we need to find (AC).

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Calculate Side Length AC in a Deltoid with Area 45 cm² and DB = 9

Deltoid Area 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 quadrilaterals?

Why do we use the diagonal formula for deltoid area?

How do I know which measurements are the diagonals?

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

Can I solve this problem without using the area formula?

🌟 Unlock Your Math Potential

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