Calculate the Area of a Deltoid with Diagonals 7 and 10 Units

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 two pairs of adjacent sides that are equal. Unlike rectangles or squares, deltoids have perpendicular diagonals, making the diagonal formula work perfectly!

Q: Why do we divide by 2 in the diagonal formula?

The diagonals of a deltoid create four right triangles inside the shape. The full product of diagonals gives the area of a rectangle, but we only want half of that area - the actual deltoid area.

Q: How can I remember which numbers are the diagonals?

Look for the lines that cross through the center of the deltoid! In this problem, the horizontal line (AC = 10) and vertical line (BD = 7) are clearly marked as the diagonals.

Q: What if the diagonals aren't perpendicular?

If the diagonals aren't perpendicular, you cannot use this formula! The $ A = \frac{1}{2} \times d_1 \times d_2 $ formula only works when diagonals meet at 90° angles, which is true for deltoids.

Q: Can I use this formula for any quadrilateral?

No! This diagonal formula only works for rhombi, squares, and deltoids (kites) - shapes where diagonals are perpendicular. For other quadrilaterals, you need different area formulas.

Area Calculation with Diagonal Measurements

Given the deltoid ABCD

Find the area

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:03 Let's find the area of the kite.

00:06 We're going to use a special formula for the area of a kite.

00:11 It's diagonal one times diagonal two, all divided by 2.

00:16 Now, let's plug in the given values and solve to find the area.

00:31 We need to divide 10 by 2.

00:40 And that's the solution to our question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the deltoid ABCD

Find the area

Step-by-step solution

To solve this problem, we need to calculate the area of the deltoid using the formula for the area in terms of diagonals:

Identify the two diagonals: $AC = 10$ cm and $BD = 7$ cm.
Use the formula for the area of a deltoid: $A = \frac{1}{2} \times d_1 \times d_2$ .
Substitute the values of the diagonals into the formula: $A = \frac{1}{2} \times 10 \times 7 = \frac{70}{2} = 35$ .

Thus, the area of the deltoid is $\textbf{35 cm}^2$ .

Therefore, the solution to the problem is $\textbf{35 cm}^2$ , which corresponds to choice 3.

Final Answer

$35$ cm².

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonals
Technique: Apply $A = \frac{1}{2} \times 10 \times 7 = 35$
Check: Verify diagonals are perpendicular and formula gives reasonable area ✓

Common Mistakes

Avoid these frequent errors

Adding diagonals instead of multiplying
Don't add the diagonals (10 + 7 = 17) thinking it gives area! This ignores the geometric relationship between diagonals and area. Always multiply the diagonals first, then divide by 2.

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 two pairs of adjacent sides that are equal. Unlike rectangles or squares, deltoids have perpendicular diagonals, making the diagonal formula work perfectly!

Why do we divide by 2 in the diagonal formula?

The diagonals of a deltoid create four right triangles inside the shape. The full product of diagonals gives the area of a rectangle, but we only want half of that area - the actual deltoid area.

How can I remember which numbers are the diagonals?

Look for the lines that cross through the center of the deltoid! In this problem, the horizontal line (AC = 10) and vertical line (BD = 7) are clearly marked as the diagonals.

What if the diagonals aren't perpendicular?

If the diagonals aren't perpendicular, you cannot use this formula! The $A = \frac{1}{2} \times d_1 \times d_2$ formula only works when diagonals meet at 90° angles, which is true for deltoids.

Can I use this formula for any quadrilateral?

No! This diagonal formula only works for rhombi, squares, and deltoids (kites) - shapes where diagonals are perpendicular. For other quadrilaterals, you need different area formulas.

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