Deltoid Geometry: Calculate CO Given AO=6cm, BO=5cm, and Area=80cm²

Q: Why is BD = 2 × BO instead of BO + OD?

In a deltoid, the diagonals are perpendicular and one diagonal (BD) is the axis of symmetry. This means BO = OD, so the full length BD = BO + OD = BO + BO = 2 × BO = 10 cm.

Q: How do I know which diagonal is the axis of symmetry?

The diagonal connecting the vertices where unequal sides meet is the axis of symmetry. Since AD = AB and CB = CD, points B and D are where unequal sides meet, making BD the symmetry axis.

Q: Can I use the triangle area method instead?

Yes! You can split the deltoid into triangles, but the diagonal formula is much faster: Area = $ \frac{1}{2} \cdot AC \cdot BD $. Just remember to use complete diagonal lengths, not segments.

Q: What if AO and CO were different from 6 and 10?

The method stays the same! You'd still use Area = $ \frac{1}{2} \cdot (AO + CO) \cdot BD $, substitute your known values, and solve for the unknown segment. The key is always using full diagonal lengths.

Deltoid Area Formula with Diagonal Segments

Given ABCD deltoid AD=AB CB=CD

The diagonals of the deltoid intersect at the point O

Given in cm AO=6 BO=5

The area of the deltoid is equal to 80 cm².

Calculate the side CO

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:15 Let's find the length of side C O.

00:19 We'll use the formula to calculate the area of a kite.

00:23 It's diagonal times diagonal, divided by 2.

00:29 The main diagonal intersects the secondary diagonal.

00:47 We'll substitute the value of B O to find B D.

00:57 Next, substitute B D into the kite area formula.

01:03 Multiply by 2 to get rid of the fraction.

01:09 Let's isolate A C.

01:12 This gives us the length of diagonal A C.

01:18 The side A C is the sum of A O and O C.

01:24 Now, plug in the side values to find O C.

01:29 And that's the solution to our problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given ABCD deltoid AD=AB CB=CD

The diagonals of the deltoid intersect at the point O

Given in cm AO=6 BO=5

The area of the deltoid is equal to 80 cm².

Calculate the side CO

Step-by-step solution

To solve for $CO$ , we will use the area formula for the deltoid:

Step 1: Calculate full length of diagonal $BD$ :

$BD = 2 \times BO = 2 \times 5 = 10 \text{ cm}$ .

Step 2: Use the kite area formula:
$\text{Area} = \frac{1}{2} \cdot AC \cdot BD$ .

Substitute known values into the formula:

$80 = \frac{1}{2} \cdot (6 + CO) \cdot 10$ .

Step 3: Simplify and solve for $CO$ :

$80 = 5 \cdot (6 + CO)$ leads to

$80 = 30 + 5CO$ .

Solving for $CO$ , we subtract 30 from both sides:

$50 = 5CO$ ,

$CO = \frac{50}{5} = 10$ .

Therefore, the side $CO$ is 10 cm.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Deltoid Properties: Diagonals are perpendicular; one diagonal is axis of symmetry
Area Formula: Area = $\frac{1}{2} \cdot d_1 \cdot d_2$ where d₁ and d₂ are full diagonal lengths
Verification: Check that AO + CO = AC and substitute back into area formula ✓

Common Mistakes

Avoid these frequent errors

Using segment lengths instead of full diagonal lengths in area formula
Don't substitute AO = 6 and BO = 5 directly into Area = ½ × 6 × 5 = 15 cm²! This uses segments, not full diagonals, giving a completely wrong area. Always find the complete diagonal lengths: AC = AO + CO and BD = 2 × BO before applying the area formula.

Practice Quiz

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FAQ

Everything you need to know about this question

Why is BD = 2 × BO instead of BO + OD?

In a deltoid, the diagonals are perpendicular and one diagonal (BD) is the axis of symmetry. This means BO = OD, so the full length BD = BO + OD = BO + BO = 2 × BO = 10 cm.

How do I know which diagonal is the axis of symmetry?

The diagonal connecting the vertices where unequal sides meet is the axis of symmetry. Since AD = AB and CB = CD, points B and D are where unequal sides meet, making BD the symmetry axis.

Can I use the triangle area method instead?

Yes! You can split the deltoid into triangles, but the diagonal formula is much faster: Area = $\frac{1}{2} \cdot AC \cdot BD$ . Just remember to use complete diagonal lengths, not segments.

What if AO and CO were different from 6 and 10?

The method stays the same! You'd still use Area = $\frac{1}{2} \cdot (AO + CO) \cdot BD$ , substitute your known values, and solve for the unknown segment. The key is always using full diagonal lengths.

How can I check my answer of CO = 10?

Substitute back: AC = AO + CO = 6 + 10 = 16 cm, BD = 10 cm. Area = $\frac{1}{2} \times 16 \times 10 = 80$ cm² ✓. This matches the given area!

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Deltoid Geometry: Calculate CO Given AO=6cm, BO=5cm, and Area=80cm²

Deltoid Area Formula with Diagonal Segments

❤️ 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

Why is BD = 2 × BO instead of BO + OD?

How do I know which diagonal is the axis of symmetry?

Can I use the triangle area method instead?

What if AO and CO were different from 6 and 10?

How can I check my answer of CO = 10?

🌟 Unlock Your Math Potential

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