Calculate the Side Length in a Deltoid: Given Area 72 cm² and BM = 2 cm

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

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. It has perpendicular diagonals that intersect, making the area formula very useful!

Q: Why do we assume M is the midpoint of diagonal BD?

In a deltoid, the diagonals are perpendicular and one bisects the other. Since M lies on diagonal BD and the figure shows symmetry, M is the intersection point where AC bisects BD.

Q: Can I use a different area formula for this problem?

While there are other area formulas, the diagonal formula $ \frac{1}{2} \times d_1 \times d_2 $ is most direct when you know information about the diagonals, like BM in this problem.

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

It doesn't matter! Since you're multiplying the diagonals, $ d_1 \times d_2 = d_2 \times d_1 $. Just make sure you identify both complete diagonal lengths correctly.

Deltoid Area Formula with Diagonal Relationships

Shown below is the deltoid ABCD.

Side length BM equals 2 cm.

The area of the deltoid is 72 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

00:00 Find AC

00:03 In a kite, the main diagonal intersects the secondary diagonal

00:13 The whole side equals the sum of its parts

00:18 We'll use the formula for calculating the area of a kite

00:22 (diagonal times diagonal) divided by 2

00:27 We'll substitute appropriate values according to the given data and solve for AC

00:34 Divide 4 by 2

00:37 Isolate AC

00:47 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

Shown below is the deltoid ABCD.

Side length BM equals 2 cm.

The area of the deltoid is 72 cm².

Find the length of the side AC.

Step-by-step solution

To solve this problem, we'll employ the formula for the area of a kite or deltoid, which relates to its diagonals AC and BD.

The formula is:

\text{Area} = \frac{1}{2} \times d1 \times d2

Given that the diagonal BD consists of BM and MD, and BM = MD as M is the midpoint, we have:

d2 = BD = BM + MD = 2 + 2 = 4 \text{ cm}

Also, the area is given as 72 cm². We substitute into the area formula:

72 = \frac{1}{2} \times AC \times 4

Simplifying the equation by multiplying through by 2 to eliminate the fraction:

144 = AC \times 4

Divide both sides by 4 to solve for AC:

AC = \frac{144}{4}

Therefore:

AC = 36 \text{ cm}

Thus, the length of side AC is $\textbf{36 cm}$ .

Final Answer

$36$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonal lengths
Technique: Since M is midpoint, BD = 2 × BM = 4 cm
Check: Substitute AC = 36: Area = ½ × 36 × 4 = 72 cm² ✓

Common Mistakes

Avoid these frequent errors

Using BM as the full diagonal length
Don't use BM = 2 cm as the diagonal BD = wrong area calculation! M is the midpoint, so BM is only half of diagonal BD. Always find the complete diagonal: BD = BM + MD = 2 + 2 = 4 cm.

Practice Quiz

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FAQ

Everything you need to know about this question

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

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. It has perpendicular diagonals that intersect, making the area formula very useful!

Why do we assume M is the midpoint of diagonal BD?

In a deltoid, the diagonals are perpendicular and one bisects the other. Since M lies on diagonal BD and the figure shows symmetry, M is the intersection point where AC bisects BD.

Can I use a different area formula for this problem?

While there are other area formulas, the diagonal formula $\frac{1}{2} \times d_1 \times d_2$ is most direct when you know information about the diagonals, like BM in this problem.

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

It doesn't matter! Since you're multiplying the diagonals, $d_1 \times d_2 = d_2 \times d_1$ . Just make sure you identify both complete diagonal lengths correctly.

What if I calculated BD wrong - how would that affect my answer?

If BD is wrong, your final answer will be wrong too! For example, if you used BD = 2 instead of 4, you'd get AC = 72 cm instead of 36 cm. Always double-check your diagonal calculations first.

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