Finding Side Length AC in a Deltoid with Area 72 cm² and MD = 3 cm

Q: What exactly is a deltoid?

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent equal sides. Its diagonals are perpendicular, and one diagonal bisects the other at right angles.

Q: Can I use a different area formula for deltoids?

The standard formula $ Area = \frac{1}{2} \times d_1 \times d_2 $ where d₁ and d₂ are the diagonal lengths is the most reliable method for deltoids.

Q: How do I find the other diagonal if I only know one half?

If you know half of one diagonal (like MD = 3), double it to get the full diagonal (BD = 6). Then use the area formula to find the other diagonal length.

Q: What if my answer doesn't match any of the given options?

Double-check your calculations! Make sure you're using the complete diagonal lengths, not just the half-lengths from the center point M.

Deltoid Area Formula with Diagonal Calculations

Given below is the deltoid ABCD.

Side length MD equals 3 cm.

The area of the deltoid is 72 cm².

What is the length of the side AC?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's find A C, the diagonal of the kite.

00:16 In a kite, the main diagonal crosses the secondary diagonal at a right angle.

00:29 Remember, each side is the sum of its segments.

00:34 We'll use the formula for the area of a kite.

00:38 Multiply both diagonals, then divide by 2.

00:44 Substitute the given values to find A C.

00:53 First, divide 6 by 2.

00:57 Next, isolate A C to solve.

01:06 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given below is the deltoid ABCD.

Side length MD equals 3 cm.

The area of the deltoid is 72 cm².

What is the length of the side AC?

Step-by-step solution

To solve for the length of $AC$ in the deltoid:

Given $MD = 3$ cm, which implies $BD = 2 \times MD = 6$ cm.
The area of the deltoid is given by the formula: $\text{Area} = \frac{1}{2} \times AC \times BD$ .

Putting the known values into the formula:
$72 = \frac{1}{2} \times AC \times 6$ .

To isolate $AC$ , multiply both sides by 2:

$144 = AC \times 6$ .

Divide both sides by 6 to solve for $AC$ :

$AC = \frac{144}{6} = 24 \, \text{cm}$ .

Therefore, the length of the side $AC$ is $24 \, \text{cm}$ .

Final Answer

$24$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonal lengths
Technique: Since MD = 3 cm, then BD = 2 × 3 = 6 cm
Check: Verify with formula: $\frac{1}{2} \times 24 \times 6 = 72$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using MD instead of the full diagonal BD
Don't use MD = 3 cm directly in the area formula = wrong answer of 108 cm! M is the intersection point, so MD is only half of diagonal BD. Always use the complete diagonal lengths: BD = 2 × MD = 6 cm.

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

Why is BD equal to 2 × MD?

In a deltoid, the diagonals intersect at point M. Since M is the center point, it divides each diagonal into two equal parts. So if MD = 3 cm, then the full diagonal BD = 3 + 3 = 6 cm.

What exactly is a deltoid?

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent equal sides. Its diagonals are perpendicular, and one diagonal bisects the other at right angles.

Can I use a different area formula for deltoids?

The standard formula $Area = \frac{1}{2} \times d_1 \times d_2$ where d₁ and d₂ are the diagonal lengths is the most reliable method for deltoids.

How do I find the other diagonal if I only know one half?

If you know half of one diagonal (like MD = 3), double it to get the full diagonal (BD = 6). Then use the area formula to find the other diagonal length.

What if my answer doesn't match any of the given options?

Double-check your calculations! Make sure you're using the complete diagonal lengths, not just the half-lengths from the center point M.

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