Deltoid Geometry: Finding AC Length Given Area 54 cm² and BD = 6 cm

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 parallelograms, its diagonals are perpendicular and only one diagonal bisects the other.

Q: Why do we use ½ × d₁ × d₂ for the area formula?

This formula works because the perpendicular diagonals divide the deltoid into 4 right triangles. The area equals the sum of these triangles, which simplifies to $ \frac{1}{2} \times d_1 \times d_2 $.

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

It doesn't matter! Since we're multiplying the diagonals, you can call either one d₁ or d₂. In this problem, BD = 6 cm and AC is what we're solving for.

Q: What if I get confused between different quadrilateral area formulas?

Remember: rectangles use length × width, parallelograms use base × height, but deltoids and rhombi use ½ × diagonal₁ × diagonal₂ when diagonals are perpendicular.

Q: Can I check my answer a different way besides substituting back?

Yes! You can also think of it as: "If the area is 54 and one diagonal is 6, the other must be 18" because 54 ÷ 3 = 18. The key insight is that $ \frac{1}{2} \times 6 = 3 $.

Area Formula with Diagonal Calculation

Given the deltoid ABCD

Side length BD equals 6 cm

The area of the deltoid is equal to 54 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

00:11 Let's find the length of A C.

00:14 We'll use the formula to calculate the area of a kite.

00:18 It's diagonal one times diagonal two, divided by two.

00:23 Now, substitute the given values and solve for A C.

00:34 First, let's divide six by two.

00:41 Next, isolate A C in the equation.

00:53 And there you have it, the answer!

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 6 cm

The area of the deltoid is equal to 54 cm².

Find the length of the side AC

Step-by-step solution

To find the length of side $AC$ , follow these steps:

Step 1: Use the formula for the area of a deltoid: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ .
Step 2: Set the known values into the equation, where $d_1 = BD = 6$ cm and the area is 54 cm $^2$ .
Step 3: Rearrange the formula to solve for $d_2$ (which is $AC$ ): $54 = \frac{1}{2} \times 6 \times AC$ .
Step 4: Simplify and solve for $AC$ : $54 = 3 \times AC$ .
Step 5: Divide both sides by 3 to isolate $AC$ : $AC = \frac{54}{3} = 18$ cm.

Therefore, the length of $AC$ is $18$ cm.

Final Answer

$18$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals one-half times product of diagonals
Technique: Substitute known values: $54 = \frac{1}{2} \times 6 \times AC$
Check: Verify $\frac{1}{2} \times 6 \times 18 = 54$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for deltoid
Don't use base × height for a deltoid = wrong answer! Deltoids aren't rectangles or triangles. Always use the diagonal formula: Area = ½ × d₁ × d₂ for quadrilaterals with perpendicular diagonals.

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 (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or parallelograms, its diagonals are perpendicular and only one diagonal bisects the other.

Why do we use ½ × d₁ × d₂ for the area formula?

This formula works because the perpendicular diagonals divide the deltoid into 4 right triangles. The area equals the sum of these triangles, which simplifies to $\frac{1}{2} \times d_1 \times d_2$ .

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

It doesn't matter! Since we're multiplying the diagonals, you can call either one d₁ or d₂. In this problem, BD = 6 cm and AC is what we're solving for.

What if I get confused between different quadrilateral area formulas?

Remember: rectangles use length × width, parallelograms use base × height, but deltoids and rhombi use ½ × diagonal₁ × diagonal₂ when diagonals are perpendicular.

Can I check my answer a different way besides substituting back?

Yes! You can also think of it as: "If the area is 54 and one diagonal is 6, the other must be 18" because 54 ÷ 3 = 18. The key insight is that $\frac{1}{2} \times 6 = 3$ .

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