Calculate Side Length AC in Deltoid with Area 48 cm² and Diagonal 6 cm

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

A deltoid is a special quadrilateral (4-sided shape) where the diagonals are perpendicular to each other. Unlike rectangles or parallelograms, deltoids use a unique area formula based on their perpendicular diagonals.

Q: Why do we multiply the diagonals and then divide by 2?

When diagonals are perpendicular, they create four right triangles inside the deltoid. The formula $ \frac{1}{2} \times d_1 \times d_2 $ calculates the total area of all four triangles combined!

Q: How do I know which diagonal is which in the problem?

It doesn't matter! The area formula works with either diagonal first. Whether you write $ \frac{1}{2} \times AC \times DB $ or $ \frac{1}{2} \times DB \times AC $, you'll get the same answer.

Q: What if I get stuck on the algebra part?

Break it down step by step: 48 = 3 × AC means "3 times what number equals 48?" Divide both sides by 3 to find AC = 16. Practice makes perfect!

Q: Can I use this formula for any quadrilateral?

No! This formula only works when the diagonals are perpendicular (meet at 90°). For other quadrilaterals like rectangles or trapezoids, you need different formulas.

Deltoid Area with Diagonal Calculations

Shown below is the deltoid ABCD.

Diagonal DB = 6

The area of the deltoid is 48 cm².

Calculate the side AC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate 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 and solve for AC

00:22 We'll multiply by 2 to eliminate the fraction

00:35 Isolate AC

00:43 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

Shown below is the deltoid ABCD.

Diagonal DB = 6

The area of the deltoid is 48 cm².

Calculate the side AC.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Use the formula for the area of a deltoid.
Step 2: Substitute the known values into the formula.
Step 3: Solve for the unknown diagonal $AC$ .

Now, let's work through each step:

Step 1: The area of a deltoid is given by the formula:

$\text{Area} = \frac{1}{2} \times AC \times DB$

Step 2: Substitute the given values into the formula:

$48 = \frac{1}{2} \times AC \times 6$

Step 3: Solve for $AC$ .

First, simplify the equation:

$48 = 3 \times AC$

Now, divide both sides by 3 to isolate $AC$ :

$AC = \frac{48}{3} = 16$

Therefore, the solution to the problem is $AC = 16 \, \text{cm}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonals
Technique: Substitute Area = 48, DB = 6 into $\frac{1}{2} \times AC \times 6$
Check: Verify $\frac{1}{2} \times 16 \times 6 = 48$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for deltoids
Don't use triangle area formulas like base × height ÷ 2 for deltoids = completely wrong answer! Deltoids are special quadrilaterals where diagonals are perpendicular. Always use the deltoid formula: Area = ½ × diagonal₁ × diagonal₂.

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 shapes?

A deltoid is a special quadrilateral (4-sided shape) where the diagonals are perpendicular to each other. Unlike rectangles or parallelograms, deltoids use a unique area formula based on their perpendicular diagonals.

Why do we multiply the diagonals and then divide by 2?

When diagonals are perpendicular, they create four right triangles inside the deltoid. The formula $\frac{1}{2} \times d_1 \times d_2$ calculates the total area of all four triangles combined!

How do I know which diagonal is which in the problem?

It doesn't matter! The area formula works with either diagonal first. Whether you write $\frac{1}{2} \times AC \times DB$ or $\frac{1}{2} \times DB \times AC$ , you'll get the same answer.

What if I get stuck on the algebra part?

Break it down step by step: 48 = 3 × AC means "3 times what number equals 48?" Divide both sides by 3 to find AC = 16. Practice makes perfect!

Can I use this formula for any quadrilateral?

No! This formula only works when the diagonals are perpendicular (meet at 90°). For other quadrilaterals like rectangles or trapezoids, you need different formulas.

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