Deltoid Area Problem: Find Variable 'a' Given Area 45 cm²

Q: Why do we multiply the diagonal segments a + 3a?

In a deltoid, the diagonal AC is made up of two segments: AM = a and MC = 3a. The total diagonal length is a + 3a = 4a, which we need for the area formula.

Q: How do I know the diagonals are perpendicular?

The diagram shows a right angle symbol where the diagonals intersect at point M. This confirms they meet at 90°, so we can use the standard deltoid area formula.

Q: What if I get a decimal answer - is that normal?

Yes! Getting $ a = 4.5 $ cm is perfectly correct. Many geometry problems have decimal solutions, especially when working with areas and measurements.

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

The formula $ \text{Area} = \frac{1}{2} \times d_1 \times d_2 $ works when diagonals are perpendicular. For non-perpendicular diagonals, you'd need a more complex formula involving the angle between them.

Deltoid Area with Perpendicular Diagonals

Shown below is the deltoid ABCD.

Side length BD equals 5 cm.

The area of the deltoid is 45 cm².

What is the the value of $a?$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find A

00:03 The entire side equals the sum of its parts

00:11 The size of diagonal BD according to the given data

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

00:18 (diagonal times diagonal) divided by 2

00:21 We'll substitute appropriate values according to the given data and solve for A

00:31 We'll isolate A

00:45 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 BD equals 5 cm.

The area of the deltoid is 45 cm².

What is the the value of $a?$

Step-by-step solution

Let's solve this problem by working through the steps:

We are given a deltoid ABCD where:

The length of diagonal BD is 5 cm.
The sum of the segments forming diagonal AC is $a + 3a = 4a$ .
The area of the deltoid is 45 cm².

We use the area formula for a deltoid when the diagonals intersect at right angles:

$\text{Area} = \frac{1}{2} \times d_1 \times d_2$

Here, $d_1 = 5$ cm and $d_2 = 4a$ . Substituting these values into the formula:

$\frac{1}{2} \times 5 \times 4a = 45$

Simplifying this equation:

$\frac{1}{2} \times 20a = 45$

$10a = 45$

Now, solve for $a$ :

$a = \frac{45}{10}$

$a = 4.5$

Therefore, the length of segment $a$ is $4.5 \, \text{cm}$ .

Final Answer

$4.5$

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonal lengths
Technique: Set up $\frac{1}{2} \times 5 \times 4a = 45$ and solve
Check: Verify $\frac{1}{2} \times 5 \times 18 = 45$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using wrong diagonal measurements
Don't add segments a + 3a = 4a and then use just 'a' as diagonal length = missing 75% of the diagonal! This gives area of 11.25 instead of 45. Always use the complete diagonal length when applying the area formula.

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 do we multiply the diagonal segments a + 3a?

In a deltoid, the diagonal AC is made up of two segments: AM = a and MC = 3a. The total diagonal length is a + 3a = 4a, which we need for the area formula.

How do I know the diagonals are perpendicular?

The diagram shows a right angle symbol where the diagonals intersect at point M. This confirms they meet at 90°, so we can use the standard deltoid area formula.

What if I get a decimal answer - is that normal?

Yes! Getting $a = 4.5$ cm is perfectly correct. Many geometry problems have decimal solutions, especially when working with areas and measurements.

Can I use a different area formula for deltoids?

The formula $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ works when diagonals are perpendicular. For non-perpendicular diagonals, you'd need a more complex formula involving the angle between them.

How do I check my work once I find a = 4.5?

Substitute back: if a = 4.5, then the diagonal AC = 4a = 4(4.5) = 18 cm. Check: $\frac{1}{2} \times 5 \times 18 = 45$ cm² ✓

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