Calculate Triangle Ratios in a Deltoid: Using the 1:5 Point Relationship

Q: How can I visualize this problem better?

Think of triangles ABD and BCD as having the same base BD. Point O on diagonal AC acts like different 'heights' for these triangles. The closer O is to A, the smaller triangle ABD becomes compared to BCD.

Q: Does this work for any deltoid shape?

Yes! This principle works for any quadrilateral where you divide a diagonal. The key is identifying which triangles share a common base and how the dividing point affects their relative areas.

Q: What's the difference between a deltoid and other quadrilaterals here?

For this problem, the deltoid shape doesn't matter! The area ratio depends only on how point O divides diagonal AC and which triangles share base BD. The same principle applies to rectangles, parallelograms, or any quadrilateral.

Area Ratios with Point Division Properties

Look the deltoid ABCD shown below.

The ratio between AO and OC is 1:5.

Calculate the ratio between triangle ABD and triangle BCD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the ratio of triangles ABD to BCD

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

00:18 We'll do the same thing for triangle BCD

00:30 We'll substitute the triangles' formulas in the ratio

00:41 We'll simplify what we can

00:53 And that's the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look the deltoid ABCD shown below.

The ratio between AO and OC is 1:5.

Calculate the ratio between triangle ABD and triangle BCD.

Step-by-step solution

To determine the area ratio between triangles $\triangle ABD$ and $\triangle BCD$ , we will compare the segments derived from the given point O.

We are given that the ratio $AO : OC = 1 : 5$ .
Both triangles $\triangle ABD$ and $\triangle BCD$ share the line segment BD as a base, with their 'perpendicular heights' being the same when considering B as the vertex and segments AO and OC as part of their respective triangles.
The areas of triangles sharing the same base and height are proportional to the lengths of the other segment partitions they connect to.
Since O divides AC in the mentioned ratio, AO is $1/6$ of AC and OC is $5/6$ of AC.

Thus, the ratio of the areas of the triangles, based on the aforementioned proportions of their respective line segments, becomes $\frac{1}{5}$ .

This simplifies to the answer of $1:5$

Therefore, the ratio of the areas of triangle $\triangle ABD$ to triangle $\triangle BCD$ is $1:5$ .

Final Answer

1:5

Key Points to Remember

Essential concepts to master this topic

Ratio Property: Triangles sharing same base have areas proportional to heights
Technique: Since AO:OC = 1:5, triangle ABD:BCD = 1:5
Check: Both triangles share BD base, so height ratio equals area ratio ✓

Common Mistakes

Avoid these frequent errors

Using the wrong triangles for comparison
Don't compare triangle areas that don't share the same base = meaningless ratios! This ignores the geometric relationship between point O and the diagonal AC. Always identify triangles that share a common base or vertex when working with area ratios.

Practice Quiz

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What is the ratio between the orange and gray parts in the drawing?

FAQ

Everything you need to know about this question

Why do triangles ABD and BCD have areas in the same ratio as AO:OC?

Both triangles share the same base BD. When triangles share a base, their area ratio equals their height ratio. Since O divides AC in ratio 1:5, the perpendicular distances from O create the same ratio for triangle heights.

What if I calculated the ratio as 1:6 instead of 1:5?

You might be thinking about the total length AC. Remember: if AO:OC = 1:5, then AO is $\frac{1}{6}$ of AC and OC is $\frac{5}{6}$ of AC, but the triangle ratio is still 1:5!

How can I visualize this problem better?

Think of triangles ABD and BCD as having the same base BD. Point O on diagonal AC acts like different 'heights' for these triangles. The closer O is to A, the smaller triangle ABD becomes compared to BCD.

Does this work for any deltoid shape?

Yes! This principle works for any quadrilateral where you divide a diagonal. The key is identifying which triangles share a common base and how the dividing point affects their relative areas.

What's the difference between a deltoid and other quadrilaterals here?

For this problem, the deltoid shape doesn't matter! The area ratio depends only on how point O divides diagonal AC and which triangles share base BD. The same principle applies to rectangles, parallelograms, or any quadrilateral.

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Calculate Triangle Ratios in a Deltoid: Using the 1:5 Point Relationship

Area Ratios with Point Division Properties

❤️ 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 triangles ABD and BCD have areas in the same ratio as AO:OC?

What if I calculated the ratio as 1:6 instead of 1:5?

How can I visualize this problem better?

Does this work for any deltoid shape?

What's the difference between a deltoid and other quadrilaterals here?

🌟 Unlock Your Math Potential

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