Calculate AO Length in Kite ABCD: Given Area 42 cm² and Diagonal 14 units

Q: What's the difference between AC and AO in this problem?

AC is the full diagonal from vertex A to vertex C, while AO is half the diagonal from vertex A to the center O. Since diagonals bisect each other in kites, AO = AC/2.

Q: Why do we use the area formula with diagonals for kites?

Kites have perpendicular diagonals that bisect each other, making the area formula $ A = \frac{d_1 \times d_2}{2} $ perfect for these shapes. It's much easier than breaking into triangles!

Q: How do I know which diagonal is which?

It doesn't matter! The area formula works with any two perpendicular diagonals. In this problem, BD = 14 is given, and we need to find AC to use the formula.

Q: Can I solve this problem differently?

Yes! You could split the kite into four right triangles and use their areas, but the diagonal method is much faster and less prone to errors.

Q: What if I got AC = 12 instead of 6?

Check your algebra! From $ 42 = \frac{AC \times 14}{2} $, multiply both sides by 2 to get $ 84 = AC \times 14 $, then divide by 14: AC = 6.

Kite Area Formula with Diagonal Calculations

The kite ABCD shown below has an area of 42 cm².

AB = BC

DC = AD

BD = 14

The diagonals of the kite intersect at point 0.

Calculate the length of side AO.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:16 Let's calculate the length of side AO.

00:19 We'll use the kite area formula.

00:22 It's diagonal one times diagonal two, divided by two.

00:28 Now, substitute the given values to find diagonal AC.

00:49 This gives us the length of diagonal AC.

00:55 Remember, the main diagonal intersects the secondary diagonal.

01:00 And that's how we solve this problem. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The kite ABCD shown below has an area of 42 cm².

AB = BC

DC = AD

BD = 14

The diagonals of the kite intersect at point 0.

Calculate the length of side AO.

Step-by-step solution

We substitute the data we have into the formula for the area of the kite:

$S=\frac{AC\times BD}{2}$

$42=\frac{AC\times14}{2}$

We multiply by 2 to remove the denominator:

$14AC=84$

Then divide by 14:

$AC=6$

In a rhombus, the main diagonal crosses the second diagonal, therefore:

$AO=\frac{AC}{2}=\frac{6}{2}=3$

Final Answer

3 cm

Key Points to Remember

Essential concepts to master this topic

Area Formula: For kites use $A = \frac{d_1 \times d_2}{2}$ where d₁, d₂ are diagonals
Technique: Solve $42 = \frac{AC \times 14}{2}$ to get AC = 6
Check: Verify AO = AC/2 since diagonals bisect: 6/2 = 3 cm ✓

Common Mistakes

Avoid these frequent errors

Using wrong formula or confusing AO with AC
Don't use AO directly in the area formula = wrong diagonal length! The area formula needs the full diagonal AC, not half of it. Always find the complete diagonal first, then divide by 2 to get AO.

Practice Quiz

Test your knowledge with interactive questions

Look at the kite ABCD below.

Diagonal DB = 10

CB = 4

Is it possible to calculate the area of the kite? If so, what is it?

FAQ

Everything you need to know about this question

What's the difference between AC and AO in this problem?

AC is the full diagonal from vertex A to vertex C, while AO is half the diagonal from vertex A to the center O. Since diagonals bisect each other in kites, AO = AC/2.

Why do we use the area formula with diagonals for kites?

Kites have perpendicular diagonals that bisect each other, making the area formula $A = \frac{d_1 \times d_2}{2}$ perfect for these shapes. It's much easier than breaking into triangles!

How do I know which diagonal is which?

It doesn't matter! The area formula works with any two perpendicular diagonals. In this problem, BD = 14 is given, and we need to find AC to use the formula.

Can I solve this problem differently?

Yes! You could split the kite into four right triangles and use their areas, but the diagonal method is much faster and less prone to errors.

What if I got AC = 12 instead of 6?

Check your algebra! From $42 = \frac{AC \times 14}{2}$ , multiply both sides by 2 to get $84 = AC \times 14$ , then divide by 14: AC = 6.

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Calculate AO Length in Kite ABCD: Given Area 42 cm² and Diagonal 14 units

Kite Area Formula with Diagonal Calculations

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What's the difference between AC and AO in this problem?

Why do we use the area formula with diagonals for kites?

How do I know which diagonal is which?

Can I solve this problem differently?

What if I got AC = 12 instead of 6?

🌟 Unlock Your Math Potential

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