Deltoid Geometry: Finding Side Length AC Given Area 60cm² and BD=15cm

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

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Its diagonals are perpendicular, which is why we can use the special area formula!

Q: How do I know which lines are the diagonals AC and BD?

Diagonals connect opposite vertices (corners) of the shape. In deltoid ABCD, AC connects A to C, and BD connects B to D. They cross inside the shape.

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

Only for shapes where diagonals are perpendicular, like deltoids, rhombuses, and squares. For rectangles or parallelograms, use base × height instead.

Deltoid Area Formula with Diagonal Calculation

Below is the deltoid ABCD.

Side length BD equals 15 cm.

The area of the deltoid is 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

00:20 We'll isolate AC

00:36 We'll divide 60 by 15

00:42 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

Below is the deltoid ABCD.

Side length BD equals 15 cm.

The area of the deltoid is 60 cm².

Find the length of the side AC.

Step-by-step solution

To solve this problem, we'll apply the following steps:

Step 1: Identify the given information: $BD = 15$ cm and the area is $60$ cm².
Step 2: Use the formula for the area of a deltoid.
Step 3: Solve for the unknown diagonal $AC$ .

Now, let's work through each step:
Step 1: We know the area formula for a deltoid is given by: $\text{Area} = \frac{1}{2} \times AC \times BD$

Step 2: Substitute the given values into the formula: $60 = \frac{1}{2} \times AC \times 15$

Step 3: Simplify and solve for $AC$ : $60 = \frac{15}{2} \times AC$
Multiply both sides by 2 to isolate $AC$ : $120 = 15 \times AC$
Divide both sides by 15: $AC = \frac{120}{15} = 8$

Therefore, the length of the side $AC$ is $8$ cm.

Final Answer

$8$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Deltoid area equals half the product of diagonal lengths
Technique: Substitute known values: $60 = \frac{1}{2} \times AC \times 15$
Check: Verify with area formula: $\frac{1}{2} \times 8 \times 15 = 60$ ✓

Common Mistakes

Avoid these frequent errors

Using perimeter or side length formulas instead of diagonal formula
Don't use regular quadrilateral area formulas that need base and height = wrong calculation! A deltoid has intersecting diagonals that are perpendicular, creating a special area relationship. Always use Area = ½ × diagonal₁ × diagonal₂ for deltoids.

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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 quadrilaterals?

A deltoid (also called a kite) is a quadrilateral with two pairs of adjacent sides that are equal. Its diagonals are perpendicular, which is why we can use the special area formula!

Why do we use ½ × diagonal₁ × diagonal₂ for the area?

When two lines cross perpendicularly, they create four right triangles. The area formula $\frac{1}{2} \times d_1 \times d_2$ adds up all these triangular areas efficiently.

How do I know which lines are the diagonals AC and BD?

Diagonals connect opposite vertices (corners) of the shape. In deltoid ABCD, AC connects A to C, and BD connects B to D. They cross inside the shape.

What if I get the diagonals mixed up in my calculation?

It doesn't matter! Since we multiply the diagonals together, $AC \times BD = BD \times AC$ . The order doesn't change the final answer.

Can I use this same formula for other quadrilaterals?

Only for shapes where diagonals are perpendicular, like deltoids, rhombuses, and squares. For rectangles or parallelograms, use base × height instead.

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