Deltoid Area Problem: Finding AC Length Given Area of 49 cm²

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

A deltoid (also called a kite) is a special quadrilateral with perpendicular diagonals. Unlike rectangles or parallelograms, deltoids use the formula Area = $ \frac{1}{2} \times d_1 \times d_2 $ where d₁ and d₂ are the diagonals.

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

The 1/2 factor comes from how deltoids are formed. When you draw both diagonals, they create four triangles. The formula $ \frac{1}{2} \times d_1 \times d_2 $ efficiently calculates the total area of all four triangles combined.

Q: How do I know which measurements are the diagonals?

Diagonals are the lines that connect opposite vertices and cross inside the shape. In this problem, AC and BD are diagonals because they connect opposite corners and intersect in the middle of the deltoid.

Q: What if I get a decimal answer instead of a whole number?

Decimal answers are perfectly valid! Just make sure to round appropriately if needed and always include the correct units (like cm, cm², etc.) in your final answer.

Q: Can I use this same formula for other quadrilaterals?

No! This formula $ \frac{1}{2} \times d_1 \times d_2 $ only works for shapes with perpendicular diagonals like deltoids, kites, and rhombuses. Rectangles and general parallelograms need different formulas.

Deltoid Area Formula with Diagonal Calculations

Given the deltoid ABCD

Side length BD equals 7 cm

The area of the deltoid is equal to 49 cm².

Find the length of the side AC

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find AC

00:03 We'll use the formula for calculating the area of a kite

00:07 (diagonal times diagonal) divided by 2

00:11 We'll substitute appropriate values according to the given data and solve for AC

00:22 We'll isolate AC

00:44 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

Given the deltoid ABCD

Side length BD equals 7 cm

The area of the deltoid is equal to 49 cm².

Find the length of the side AC

Step-by-step solution

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

Step 1: Identify the formula for the area, which is given by:
$\text{Area} = \frac{1}{2} \times BD \times AC$ where $BD$ and $AC$ are the diagonals.
Step 2: Substitute known values into the formula:
Given that $BD = 7\, \text{cm}$ and $\text{Area} = 49\, \text{cm}^2$ , we have:
$49 = \frac{1}{2} \times 7 \times AC$
Step 3: Solve the equation for $AC$ :
Multiply both sides of the equation by 2 to eliminate the fraction:
$98 = 7 \times AC$ Divide both sides by 7:
$AC = \frac{98}{7} = 14$

Therefore, the length of the side $AC$ is $14 \, \text{cm}$ .

Final Answer

$14$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals one-half times diagonal products
Technique: Substitute known values: 49 = (1/2) × 7 × AC
Check: Verify: (1/2) × 7 × 14 = 49 cm² ✓

Common Mistakes

Avoid these frequent errors

Using perimeter formula instead of area formula
Don't add the diagonal lengths BD + AC = wrong approach! This calculates perimeter, not area, giving completely incorrect results. Always use Area = (1/2) × diagonal₁ × diagonal₂ for deltoid area problems.

Practice Quiz

Test your knowledge with interactive questions

Indicate the correct answer

The next quadrilateral is:

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 special quadrilateral with perpendicular diagonals. Unlike rectangles or parallelograms, deltoids use the formula Area = $\frac{1}{2} \times d_1 \times d_2$ where d₁ and d₂ are the diagonals.

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

The 1/2 factor comes from how deltoids are formed. When you draw both diagonals, they create four triangles. The formula $\frac{1}{2} \times d_1 \times d_2$ efficiently calculates the total area of all four triangles combined.

How do I know which measurements are the diagonals?

Diagonals are the lines that connect opposite vertices and cross inside the shape. In this problem, AC and BD are diagonals because they connect opposite corners and intersect in the middle of the deltoid.

What if I get a decimal answer instead of a whole number?

Decimal answers are perfectly valid! Just make sure to round appropriately if needed and always include the correct units (like cm, cm², etc.) in your final answer.

Can I use this same formula for other quadrilaterals?

No! This formula $\frac{1}{2} \times d_1 \times d_2$ only works for shapes with perpendicular diagonals like deltoids, kites, and rhombuses. Rectangles and general parallelograms need different formulas.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Deltoid questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime