Area of Deltoid: Calculate Using 7 and 5 Unit Diagonals

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

A deltoid (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or squares, its diagonals are perpendicular but only one diagonal bisects the other, making our area formula work perfectly!

Q: Why do we multiply the diagonals and then divide by 2?

The diagonals divide the deltoid into four right triangles. When you multiply the full diagonal lengths, you get the area of a rectangle. Dividing by 2 gives you exactly half - which equals the deltoid's area!

Q: Do the diagonals have to be perpendicular for this formula to work?

Yes! This formula $ A = \frac{1}{2} \times d_1 \times d_2 $ only works when diagonals are perpendicular. Fortunately, deltoid diagonals are always perpendicular by definition!

Q: What if I can't see the diagonal measurements clearly in the diagram?

Look for colored lines and numbers in the diagram. In this problem, the red vertical line shows 7 units and the green horizontal line shows 5 units - these are your diagonals!

Q: How can I double-check my calculation of 17.5 cm²?

Try this: $ \frac{1}{2} \times 7 \times 5 = \frac{35}{2} = 17.5 $. You can also think of it as half of 35, which clearly equals 17.5 cm².

Deltoid Area with Perpendicular Diagonals

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

00:00 Find the area of the kite

00:03 We'll use the formula to calculate the area of the kite

00:07 (diagonal times diagonal) divided by 2

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

00:27 And this is the solution to the problem

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 $ABCD$ using the given lengths of its diagonals. The formula for the area of a deltoid (kite) is:

$A = \frac{1}{2} \times d_1 \times d_2$

Where $d_1$ and $d_2$ are the lengths of the diagonals. From the diagram, we know:

Diagonal $AC = 7$ cm
Diagonal $BD = 5$ cm

Substituting these values into the formula, we have:

$A = \frac{1}{2} \times 7 \times 5$

Calculating this gives:

$A = \frac{1}{2} \times 35 = 17.5$

Therefore, the area of the deltoid $ABCD$ is $17.5$ cm².

The correct answer from the given choices is:

$17.5$ cm².

Final Answer

$17.5$ cm².

Key Points to Remember

Essential concepts to master this topic

Formula: Area of deltoid equals half the product of diagonal lengths
Calculation: $A = \frac{1}{2} \times 7 \times 5 = 17.5$ cm²
Check: Verify diagonal measurements match diagram labels: AC = 7, BD = 5 ✓

Common Mistakes

Avoid these frequent errors

Adding diagonals instead of multiplying
Don't calculate 7 + 5 = 12 cm²! This gives the perimeter of the diagonals, not the area. The diagonals create triangular sections that need multiplication. Always use the formula $A = \frac{1}{2} \times d_1 \times d_2$ for deltoid area.

Practice Quiz

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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 (or kite) is a quadrilateral with two pairs of adjacent sides that are equal. Unlike rectangles or squares, its diagonals are perpendicular but only one diagonal bisects the other, making our area formula work perfectly!

Why do we multiply the diagonals and then divide by 2?

The diagonals divide the deltoid into four right triangles. When you multiply the full diagonal lengths, you get the area of a rectangle. Dividing by 2 gives you exactly half - which equals the deltoid's area!

Do the diagonals have to be perpendicular for this formula to work?

Yes! This formula $A = \frac{1}{2} \times d_1 \times d_2$ only works when diagonals are perpendicular. Fortunately, deltoid diagonals are always perpendicular by definition!

What if I can't see the diagonal measurements clearly in the diagram?

Look for colored lines and numbers in the diagram. In this problem, the red vertical line shows 7 units and the green horizontal line shows 5 units - these are your diagonals!

How can I double-check my calculation of 17.5 cm²?

Try this: $\frac{1}{2} \times 7 \times 5 = \frac{35}{2} = 17.5$ . You can also think of it as half of 35, which clearly equals 17.5 cm².

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