Deltoid Geometry: Finding AC Length Given Area 20cm² and BD = 4cm

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

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or squares, deltoids have diagonals that are perpendicular to each other, which is why we can use the simple area formula.

Q: Why do we use the diagonal lengths instead of the side lengths for area?

The deltoid's special property is that its diagonals are perpendicular (meet at 90°). This creates a natural way to calculate area using $ \text{Area} = \frac{1}{2} \times d_1 \times d_2 $, just like finding the area of a rectangle and dividing by 2.

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

Diagonals connect opposite vertices (corners). In deltoid ABCD, the diagonals are AC (from A to C) and BD (from B to D). These lines cross inside the shape and are perpendicular to each other.

Q: Can I use this same formula for any kite-shaped figure?

Yes! Any kite (deltoid) with perpendicular diagonals uses this formula. Just remember: $ \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 $

Deltoid Area Formula with Diagonal Relationships

Given the deltoid ABCD

Side length BD equals 4 cm

The area of the deltoid is equal to 20 cm².

Find the length of the side AC

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find AC

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 AC

00:23 Divide 4 by 2

00:27 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

Given the deltoid ABCD

Side length BD equals 4 cm

The area of the deltoid is equal to 20 cm².

Find the length of the side AC

Step-by-step solution

To solve for the length of side $AC$ in the deltoid $ABCD$ , we will use the deltoid area formula:

The formula for the area of a deltoid is given by $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ , where $d_1$ and $d_2$ are the lengths of the diagonals.

Given:

The area of the deltoid: $\text{Area} = 20 \, \text{cm}^2$
The length of diagonal $BD = 4 \, \text{cm}$ .
We need to find the length of diagonal $AC$ .

Substitute the known values into the formula:

$20 = \frac{1}{2} \times AC \times 4$

Re-arrange the equation to solve for $AC$ :

$20 = 2 \times AC$

Divide both sides by 2:

$AC = \frac{20}{2} = 10 \, \text{cm}$

Thus, the length of side $AC$ is $10 \, \text{cm}$ .

The only choice matching this calculation is:

$10$ cm

Final Answer

$10$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Rearrange $20 = \frac{1}{2} \times AC \times 4$ to solve for AC
Check: Verify $\frac{1}{2} \times 10 \times 4 = 20$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using perimeter formula instead of area formula
Don't add up the sides to find area = wrong measurement type! The deltoid area formula specifically uses the two diagonals multiplied together, not the side lengths. Always use $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ for deltoids.

Practice Quiz

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Indicate the correct answer

The next quadrilateral is:

FAQ

Everything you need to know about this question

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

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or squares, deltoids have diagonals that are perpendicular to each other, which is why we can use the simple area formula.

Why do we use the diagonal lengths instead of the side lengths for area?

The deltoid's special property is that its diagonals are perpendicular (meet at 90°). This creates a natural way to calculate area using $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ , just like finding the area of a rectangle and dividing by 2.

How do I know which measurements are the diagonals?

Diagonals connect opposite vertices (corners). In deltoid ABCD, the diagonals are AC (from A to C) and BD (from B to D). These lines cross inside the shape and are perpendicular to each other.

What if I'm given different information like side lengths?

If you're given side lengths instead of diagonals, you'll need additional information or use coordinate geometry to find the diagonal lengths first. The area formula always requires the diagonals for deltoids.

Can I use this same formula for any kite-shaped figure?

Yes! Any kite (deltoid) with perpendicular diagonals uses this formula. Just remember: $\text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2$

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