Deltoid Geometry: Finding Side Length BD Given Area 64 cm² and AC = 8 cm

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

A deltoid is a special quadrilateral where two pairs of adjacent sides are equal. It looks like a kite shape! Unlike rectangles or parallelograms, deltoids have perpendicular diagonals, which makes the area formula unique.

Q: Why do we use ½ × diagonal₁ × diagonal₂ for the area?

The diagonals of a deltoid are perpendicular (they meet at 90°), dividing it into four right triangles. The formula $ \frac{1}{2} \times d_1 \times d_2 $ calculates the total area of these triangles efficiently.

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

It doesn't matter! Since multiplication is commutative, AC × BD = BD × AC. Whether you call AC diagonal₁ or diagonal₂, you'll get the same answer.

Q: What if I forget to multiply both sides by 2 when solving?

You'll get half the correct answer! Always remember to eliminate fractions first. From $ 64 = \frac{1}{2} \times 8 \times BD $, multiply both sides by 2 to get $ 128 = 8 \times BD $.

Q: Can I use this same method for other kite-shaped problems?

Absolutely! This diagonal formula works for any quadrilateral with perpendicular diagonals, including rhombi and squares. Just remember: perpendicular diagonals = use this formula.

Q: How can I double-check my calculation steps?

Work backwards! Start with BD = 16, then calculate: $ \frac{1}{2} \times 8 \times 16 = 4 \times 16 = 64 $. If you get the original area, your answer is correct!

Deltoid Area Formula with Diagonal Calculations

Given the deltoid ABCD

Side length AC equals 8 cm

The area of the deltoid is equal to 64 cm².

Find the length of the side BD

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find BD

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

00:07 (diagonal times diagonal) divided by 2

00:12 We will substitute appropriate values according to the given data and solve for BD

00:22 Divide 8 by 2

00:28 Isolate BD

00:38 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 AC equals 8 cm

The area of the deltoid is equal to 64 cm².

Find the length of the side BD

Step-by-step solution

To solve the problem of finding the length of the diagonal $BD$ in deltoid $ABCD$ , where $AC = 8 \, \text{cm}$ and the area $S = 64 \, \text{cm}^2$ , follow these steps:

Step 1: Identify the given values: $AC = 8 \, \text{cm}$ and $S = 64 \, \text{cm}^2$ .
Step 2: Apply the area formula for a deltoid: $S = \frac{1}{2} \times AC \times BD$ .
Step 3: Substitute the known values into the formula.
Step 4: Solve for $BD$ .

Now, let's work through the calculation:

Given the formula for the area of a deltoid: $S = \frac{1}{2} \times AC \times BD$

Substitute the known values: $64 = \frac{1}{2} \times 8 \times BD$

To solve for $BD$ , first multiply both sides by 2 to get rid of the fraction: $128 = 8 \times BD$

Now, divide both sides by 8 to isolate $BD$ : $BD = \frac{128}{8} = 16$

Therefore, the length of $BD$ is $\textbf{16 cm}$ .

Final Answer

$16$ cm

Key Points to Remember

Essential concepts to master this topic

Area Formula: For deltoids, Area = $\frac{1}{2} \times d_1 \times d_2$
Substitution: $64 = \frac{1}{2} \times 8 \times BD$ becomes $128 = 8 \times BD$
Verification: Check: $\frac{1}{2} \times 8 \times 16 = 64$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for deltoids
Don't use base × height or side length formulas = wrong answer! Deltoids aren't regular polygons, so standard formulas don't apply. Always use the diagonal 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 where two pairs of adjacent sides are equal. It looks like a kite shape! Unlike rectangles or parallelograms, deltoids have perpendicular diagonals, which makes the area formula unique.

Why do we use ½ × diagonal₁ × diagonal₂ for the area?

The diagonals of a deltoid are perpendicular (they meet at 90°), dividing it into four right triangles. The formula $\frac{1}{2} \times d_1 \times d_2$ calculates the total area of these triangles efficiently.

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

It doesn't matter! Since multiplication is commutative, AC × BD = BD × AC. Whether you call AC diagonal₁ or diagonal₂, you'll get the same answer.

What if I forget to multiply both sides by 2 when solving?

You'll get half the correct answer! Always remember to eliminate fractions first. From $64 = \frac{1}{2} \times 8 \times BD$ , multiply both sides by 2 to get $128 = 8 \times BD$ .

Can I use this same method for other kite-shaped problems?

Absolutely! This diagonal formula works for any quadrilateral with perpendicular diagonals, including rhombi and squares. Just remember: perpendicular diagonals = use this formula.

How can I double-check my calculation steps?

Work backwards! Start with BD = 16, then calculate: $\frac{1}{2} \times 8 \times 16 = 4 \times 16 = 64$ . If you get the original area, your answer is correct!

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