Deltoid Geometry: Finding AC Length Given Area 60 cm² and BD = 12 cm

Q: What's the difference between a deltoid and a kite?

A deltoid is actually another name for a kite! Both are quadrilaterals with two pairs of adjacent sides that are equal in length.

Q: How do I identify the diagonals in a deltoid?

The diagonals are the lines that connect opposite vertices. In deltoid ABCD, the diagonals are AC and BD - they cross each other at right angles inside the shape.

Q: Why does the area formula use diagonals and not sides?

The diagonal formula $ S = \frac{1}{2} \times d_1 \times d_2 $ works because the diagonals of a kite are perpendicular. This creates four right triangles that are easy to calculate together.

Q: Can I use this formula for other quadrilaterals?

Only for kites and rhombuses! This formula specifically requires perpendicular diagonals. For rectangles, parallelograms, or trapezoids, you need different area formulas.

Deltoid Area Formula with Diagonal Calculations

Given the deltoid ABCD

Side length BD equals 12 cm

The area of the deltoid is equal to 60 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 We'll use the formula for calculating the area of a kite

00:07 (diagonal times diagonal) divided by 2

00:12 We'll substitute appropriate values according to the given data and solve for AC

00:21 Divide 12 by 2

00:25 Isolate AC

00:34 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 12 cm

The area of the deltoid is equal to 60 cm².

Find the length of the side AC

Step-by-step solution

To find the length of side $AC$ in the given deltoid:

Step 1: Write down the area formula for a kite: $S = \frac{1}{2} \times BD \times AC$ , where $S$ is the area, and $BD$ and $AC$ are the diagonals.
Step 2: Plug in the known values: $60 = \frac{1}{2} \times 12 \times AC$ .
Step 3: Simplify the equation: $60 = 6 \times AC$ .
Step 4: Solve for $AC$ :

$AC = \frac{60}{6} = 10$ cm.

Therefore, the length of side $AC$ is $10$ cm.

Final Answer

$10$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonals
Technique: Use $S = \frac{1}{2} \times BD \times AC$ where S = 60
Check: Verify $\frac{1}{2} \times 12 \times 10 = 60$ cm² ✓

Common Mistakes

Avoid these frequent errors

Using side lengths instead of diagonals in area formula
Don't use the sides of the deltoid in the area formula = wrong calculation! The formula requires the diagonals (AC and BD), not the perimeter sides. Always identify which lines are diagonals before applying the 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

What's the difference between a deltoid and a kite?

A deltoid is actually another name for a kite! Both are quadrilaterals with two pairs of adjacent sides that are equal in length.

How do I identify the diagonals in a deltoid?

The diagonals are the lines that connect opposite vertices. In deltoid ABCD, the diagonals are AC and BD - they cross each other at right angles inside the shape.

Why does the area formula use diagonals and not sides?

The diagonal formula $S = \frac{1}{2} \times d_1 \times d_2$ works because the diagonals of a kite are perpendicular. This creates four right triangles that are easy to calculate together.

What if I'm given different information like side lengths?

If you have side lengths instead of diagonals, you'll need to use a different approach. The diagonal method only works when you know at least one diagonal and the area, or both diagonals.

Can I use this formula for other quadrilaterals?

Only for kites and rhombuses! This formula specifically requires perpendicular diagonals. For rectangles, parallelograms, or trapezoids, you need different area formulas.

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