Deltoid Geometry: Calculate OC Given AO = 3cm and Triangle Ratio 1:3

Q: Why do triangle areas relate to segments AO and OC?

In a deltoid, when triangles ABD and BDC share the same base BD, their areas depend only on their heights. Since AO and OC are the perpendicular distances from A and C to line BD, the area ratio equals the height ratio!

Q: How do I set up the proportion correctly?

Since $ \frac{\text{Area ABD}}{\text{Area BDC}} = \frac{1}{3} $, we get $ \frac{AO}{OC} = \frac{1}{3} $. Cross-multiply: 1 × OC = 3 × AO, so OC = 3 × 3 = 9cm.

Q: What makes this shape a deltoid?

A deltoid is a quadrilateral with two pairs of adjacent sides that are equal. The diagonals intersect at right angles, creating the triangles we're comparing in this problem.

Deltoid Area Ratios with Proportional Segments

Shown below is the deltoid ABCD.

The ratio between triangle ABD and triangle BDC is 1:3.

Given in cm:

AO = 3

Calculate side OC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find the length of OC.

00:14 First, we'll use the formula to calculate the area of triangle A, B, D.

00:24 Next, we'll do the same for triangle B, D, C.

00:31 Now, let's substitute these area formulas into the ratio given in the problem.

00:41 We'll simplify everything where possible.

00:52 Using the information from the problem, we know the size of A, O.

00:57 We'll use this to find the value of C, O.

01:02 And that's how we solve this problem! Great job working through it!

Step-by-step written solution

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Understand the problem

Shown below is the deltoid ABCD.

The ratio between triangle ABD and triangle BDC is 1:3.

Given in cm:

AO = 3

Calculate side OC.

Step-by-step solution

To solve for $OC$ , follow these steps:

Step 1: Establish the area relationships. Given $\frac{\text{Area of } \triangle ABD}{\text{Area of } \triangle BDC} = \frac{1}{3}$ , these areas are proportional to segments $AO$ and $OC$ under height implications.
Step 2: Apply the area ratio property. Since areas are directly proportional to their corresponding triangle heights from a shared vertex, $\frac{AO}{OC} = \frac{1}{3}$ .
Step 3: Algebraically solve for $OC$ . Expressing proportions, $\frac{1}{3} \times OC = AO$ , gives $OC = 3 \times 3$ .
Step 4: Solve: $OC = 9 \, \text{cm}$ .

The accurate solution to the problem is $OC = 9 \, \text{cm}$ .

Final Answer

9 cm

Key Points to Remember

Essential concepts to master this topic

Property: Triangle areas sharing same base are proportional to their heights
Technique: If area ratio is 1:3, then segment ratio AO:OC = 1:3
Check: Verify $\frac{3}{9} = \frac{1}{3}$ matches the given area ratio ✓

Common Mistakes

Avoid these frequent errors

Confusing area ratio with segment ratio
Don't think if area ratio is 1:3 then OC = 1cm! This ignores that areas are proportional to heights when bases are equal. Always remember: if triangle area ratio is 1:3, then corresponding height ratio is also 1:3, so OC = 3 × 3 = 9cm.

Practice Quiz

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Indicate the correct answer

The next quadrilateral is:

FAQ

Everything you need to know about this question

Why do triangle areas relate to segments AO and OC?

In a deltoid, when triangles ABD and BDC share the same base BD, their areas depend only on their heights. Since AO and OC are the perpendicular distances from A and C to line BD, the area ratio equals the height ratio!

How do I set up the proportion correctly?

Since $\frac{\text{Area ABD}}{\text{Area BDC}} = \frac{1}{3}$ , we get $\frac{AO}{OC} = \frac{1}{3}$ . Cross-multiply: 1 × OC = 3 × AO, so OC = 3 × 3 = 9cm.

What makes this shape a deltoid?

A deltoid is a quadrilateral with two pairs of adjacent sides that are equal. The diagonals intersect at right angles, creating the triangles we're comparing in this problem.

Can I solve this without using ratios?

Ratios are the most direct method here! You could use coordinate geometry or trigonometry, but the proportional relationship between areas and heights makes ratios much simpler and less error-prone.

What if I got OC = 1cm instead?

That's a common error! You likely inverted the ratio. Remember: if area ratio is 1:3, the smaller triangle has area 1 and corresponds to the shorter segment AO = 3cm, so OC must be the longer segment.

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Deltoid Geometry: Calculate OC Given AO = 3cm and Triangle Ratio 1:3

Deltoid Area Ratios with Proportional 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 do triangle areas relate to segments AO and OC?

How do I set up the proportion correctly?

What makes this shape a deltoid?

Can I solve this without using ratios?

What if I got OC = 1cm instead?

🌟 Unlock Your Math Potential

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