Deltoid Geometry: Find BD Length Given AC=9cm and Area=72cm²

Q: What makes a deltoid different from other quadrilaterals?

A deltoid (kite) has two pairs of adjacent sides that are equal, and its diagonals are perpendicular. This perpendicular property is why we can use the formula $ Area = \frac{1}{2} \times d_1 \times d_2 $.

Q: Why do we use 1/2 in the deltoid area formula?

The diagonals of a deltoid are perpendicular and divide it into four right triangles. The formula $ \frac{1}{2} \times AC \times BD $ calculates the total area of these triangles.

Deltoid Area Formula with Diagonal Calculation

Given the deltoid ABCD

Side length AC equals 9 cm

The area of the deltoid is equal to 72 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

00:13 Let's find the length of B D.

00:20 Use the formula: diagonal times diagonal, divided by 2.

00:25 Now substitute the given values, and let's solve for B D.

00:36 Next, we need to isolate B D.

00:55 Great job! That's how we find the solution to this problem.

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 9 cm

The area of the deltoid is equal to 72 cm².

Find the length of the side BD

Step-by-step solution

To solve this problem, we will compute the length of diagonal BD using the formula for the area of a deltoid:

Step 1: Recall the formula for the area of a deltoid, which is given by:

\text{Area} = \frac{1}{2} \times AC \times BD

Step 2: Substitute the known values into the formula:

72 = \frac{1}{2} \times 9 \times BD

Step 3: Solve the equation for BD:

First, multiply both sides of the equation by 2 to clear the fraction:

144 = 9 \times BD

Next, divide both sides by 9 to isolate BD:

BD = \frac{144}{9}

Step 4: Perform the division:

BD = 16

Thus, the length of diagonal BD is $16$ cm.

This conclusion matches the possible answer choice 4:

The correct choice is (4): $16$ cm.

Final Answer

$16$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Substitute known values: $72 = \frac{1}{2} \times 9 \times BD$
Check: Verify: $\frac{1}{2} \times 9 \times 16 = 72$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply by 1/2 in area formula
Don't use Area = AC × BD without the 1/2 factor = double the correct area! This gives BD = 8 instead of 16. Always remember deltoid area is half the product of diagonals, just like rhombus area formula.

Practice Quiz

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FAQ

Everything you need to know about this question

What makes a deltoid different from other quadrilaterals?

A deltoid (kite) has two pairs of adjacent sides that are equal, and its diagonals are perpendicular. This perpendicular property is why we can use the formula $Area = \frac{1}{2} \times d_1 \times d_2$ .

Why do we use 1/2 in the deltoid area formula?

The diagonals of a deltoid are perpendicular and divide it into four right triangles. The formula $\frac{1}{2} \times AC \times BD$ calculates the total area of these triangles.

Can I solve this problem differently?

You could break the deltoid into triangles and find their areas separately, but using the diagonal formula is much faster and less error-prone for this type of problem.

What if I get a decimal answer for BD?

In this problem, BD works out to exactly 16 cm. But if you got a decimal, that would be fine too! Just make sure your arithmetic is correct and always check by substituting back.

How do I remember which diagonal is which?

It doesn't matter! The area formula works the same whether you call them AC and BD or d₁ and d₂. The key point is that both diagonals multiply together with the $\frac{1}{2}$ factor.

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