Deltoid Geometry: Finding Side Length BD Given Area 40 cm² and AC = 10 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 squares, its diagonals are perpendicular and only one diagonal bisects the other!

Q: Why do we use the diagonal formula instead of base times height?

For deltoids, the diagonals are the key measurements because they're perpendicular to each other. The formula $ S = \frac{1}{2} \cdot d_1 \cdot d_2 $ directly uses this special property!

Q: How do I know which lines are the diagonals AC and BD?

Diagonals connect opposite vertices of the quadrilateral. In deltoid ABCD, AC connects A to C (opposite corners) and BD connects B to D (the other pair of opposite corners).

Q: What if I accidentally switched the diagonal values in my calculation?

No problem! Since multiplication is commutative, $ \frac{1}{2} \cdot 10 \cdot 8 $ equals $ \frac{1}{2} \cdot 8 \cdot 10 $. The order doesn't matter for the final answer.

Q: Can I solve this problem if I'm only given one diagonal and the area?

Absolutely! Just rearrange the formula: if you know area and one diagonal, solve for the unknown diagonal by $ d_2 = \frac{2S}{d_1} $.

Deltoid Area Calculation with Diagonal Intersection

Given the deltoid ABCD

Side length AC equals 10 cm

The area of the deltoid is equal to 40 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

00:12 Let's find the length of B D.

00:15 We'll use the area formula for a kite.

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

00:23 Now, let's plug in the numbers we have and solve for B D.

00:33 First, divide ten by two.

00:37 Next, isolate B D in the equation.

00:48 And that's how we find B D.

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 10 cm

The area of the deltoid is equal to 40 cm².

Find the length of the side BD

Step-by-step solution

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

Step 1: Utilize the formula for the area of a kite or deltoid, $S = \frac{1}{2} \cdot d_1 \cdot d_2$ .
Step 2: Substituting the known values into this formula.
Step 3: Solve for the unknown diagonal $BD$ .

Now, let's work through each step:
Step 1: The given side $AC$ acts as the first diagonal $d_1 = 10$ cm. The area $S = 40$ cm².
Step 2: Plug these values into the formula $S = \frac{1}{2} \cdot d_1 \cdot d_2$ which becomes $40 = \frac{1}{2} \cdot 10 \cdot BD$ .
Step 3: Solving for $BD$ involves rearranging the equation: $40 = \frac{1}{2} \cdot 10 \cdot BD \implies 40 = 5 \cdot BD \implies BD = \frac{40}{5} = 8 \text{ cm}$

Therefore, the length of the side $BD$ is $8 \text{ cm}$ .

Final Answer

$8$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of both diagonals
Technique: Use $S = \frac{1}{2} \cdot d_1 \cdot d_2$ where S = 40, AC = 10
Check: Verify $\frac{1}{2} \cdot 10 \cdot 8 = 40$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using perimeter formula instead of diagonal formula
Don't add the sides like AC + BD = perimeter! This gives 18 cm instead of area. The deltoid area uses diagonal multiplication, not addition. Always use $S = \frac{1}{2} \cdot d_1 \cdot d_2$ for deltoid area calculations.

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 squares, its diagonals are perpendicular and only one diagonal bisects the other!

Why do we use the diagonal formula instead of base times height?

For deltoids, the diagonals are the key measurements because they're perpendicular to each other. The formula $S = \frac{1}{2} \cdot d_1 \cdot d_2$ directly uses this special property!

How do I know which lines are the diagonals AC and BD?

Diagonals connect opposite vertices of the quadrilateral. In deltoid ABCD, AC connects A to C (opposite corners) and BD connects B to D (the other pair of opposite corners).

What if I accidentally switched the diagonal values in my calculation?

No problem! Since multiplication is commutative, $\frac{1}{2} \cdot 10 \cdot 8$ equals $\frac{1}{2} \cdot 8 \cdot 10$ . The order doesn't matter for the final answer.

Can I solve this problem if I'm only given one diagonal and the area?

Absolutely! Just rearrange the formula: if you know area and one diagonal, solve for the unknown diagonal by $d_2 = \frac{2S}{d_1}$ .

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