Calculate the Area of Deltoid ABCD with Diagonals 8 and 9

Q: What makes a deltoid different from other quadrilaterals?

A deltoid (or kite) has two pairs of adjacent sides that are equal in length. Most importantly, its diagonals are always perpendicular, which is why we can use this simple area formula!

Q: Why do we divide by 2 in the area formula?

The diagonals split the deltoid into 4 right triangles. When you multiply the full diagonal lengths, you're calculating the area of a rectangle. Dividing by 2 gives you the actual deltoid area inside that rectangle.

Q: Do the diagonals have to intersect at their midpoints?

No! Unlike a rhombus, deltoid diagonals don't always bisect each other. But they are always perpendicular, which is what matters for the area formula.

Q: What if I'm given side lengths instead of diagonals?

You'll need to use other methods like coordinate geometry or break the deltoid into triangles. The diagonal formula only works when you know both diagonal lengths.

Q: How can I remember this area formula?

Think: "Half the diagonals dancing" - take half of the product when the diagonals are perpendicular (dancing at right angles)!

Deltoid Area with Perpendicular Diagonals

Given the deltoid ABCD

Find the area

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of the kite

00:03 We'll use the formula to calculate the kite's area

00:07 (diagonal times diagonal) divided by 2

00:17 We'll substitute appropriate values according to the given data and solve for the area

00:25 Divide 8 by 2

00:33 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

Find the area

Step-by-step solution

To solve this problem, we'll calculate the area of the deltoid $ABCD$ using the formula for the area of a kite or deltoid, which depends on its diagonals.

Step 1: Identify the given information
The given diagonals are $AC = 9$ cm and $BD = 8$ cm.
Step 2: Apply the area formula for a deltoid
The area $A$ of a deltoid with perpendicular diagonals is given by:
$A = \frac{1}{2} \times d_1 \times d_2$
Step 3: Perform the calculation
Substitute the given diagonal lengths into the formula:
$A = \frac{1}{2} \times 9 \times 8$
$A = \frac{1}{2} \times 72$
$A = 36$

Thus, the area of the deltoid $ABCD$ is $36$ cm².

Final Answer

$36$ cm².

Key Points to Remember

Essential concepts to master this topic

Formula: Area equals half the product of diagonal lengths
Technique: $A = \frac{1}{2} \times 9 \times 8 = 36$ cm²
Check: Verify diagonals are perpendicular in deltoid shapes ✓

Common Mistakes

Avoid these frequent errors

Adding diagonal lengths instead of multiplying
Don't add the diagonals (9 + 8 = 17) and divide by 2 = wrong area of 8.5! This gives the average length, not area. Always multiply the diagonals first, then divide by 2 for the correct area formula.

Practice Quiz

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Indicate the correct answer

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FAQ

Everything you need to know about this question

What makes a deltoid different from other quadrilaterals?

A deltoid (or kite) has two pairs of adjacent sides that are equal in length. Most importantly, its diagonals are always perpendicular, which is why we can use this simple area formula!

Why do we divide by 2 in the area formula?

The diagonals split the deltoid into 4 right triangles. When you multiply the full diagonal lengths, you're calculating the area of a rectangle. Dividing by 2 gives you the actual deltoid area inside that rectangle.

Do the diagonals have to intersect at their midpoints?

No! Unlike a rhombus, deltoid diagonals don't always bisect each other. But they are always perpendicular, which is what matters for the area formula.

What if I'm given side lengths instead of diagonals?

You'll need to use other methods like coordinate geometry or break the deltoid into triangles. The diagonal formula only works when you know both diagonal lengths.

How can I remember this area formula?

Think: "Half the diagonals dancing" - take half of the product when the diagonals are perpendicular (dancing at right angles)!

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