The points A, B, C and D all lie on the same line segment, in that order, such that the ratio of A, B, colon, B, C, colon, C, D*AB*:*BC*:*CD* is equal to 2:1:6. If A, D, equals, 36, *AD*=36, find B, D, .*BD*.

Given that points \( A \), \( B \), \( C \), and \( D \) lie on the same line segment in that order, and the ratio \( AB:BC:CD = 2:1:6 \), with \( AD = 36 \), we are to find the length of \( BD \).

### Step 1: Assigning Variables
Let the length of \( AB = 2x \), \( BC = x \), and \( CD = 6x \). The total length \( AD \) is then given by the sum of these segments:
\[
AD = AB + BC + CD = 2x + x + 6x = 9x
\]

### Step 2: Solving for \( x \)
We are given that \( AD = 36 \). Therefore:
\[
9x = 36
\]
\[
x = \frac{36}{9} = 4
\]

### Step 3: Finding \( BD \)
The length \( BD \) is the sum of \( BC \) and \( CD \):
\[
BD = BC + CD = x + 6x = 7x
\]
Substituting \( x = 4 \) into the equation:
\[
BD = 7(4) = 28
\]

### Final Answer
The length of \( BD \) is \( \boxed{28} \).

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. If \( AB \) were doubled while keeping \( BC \) and \( CD \) the same, what would \( BD \) be?
2. What would be the length of \( BD \) if \( AD \) was 45 instead of 36?
3. How would the calculation change if the ratio \( AB:BC:CD \) was 3:2:7?
4. If the ratio was expressed differently, such as \( 4:2:12 \), what would \( BD \) be?
5. How would you find \( BD \) if only given \( AB \) and \( CD \), with no total length provided?

**Tip:** When working with ratios in line segments, assigning a variable to the smallest segment often simplifies calculations.

The points A, B, C, and D all lie on the same line segment, in that order, such that the ratio of AB:BC:CD is equal to 2:1:6. If AD = 36, find BD.

Learn how to solve a geometry problem involving ratios on a line segment. Given points A, B, C, and D with ratios AB:BC:CD = 2:1:6 and AD = 36, find the length of BD. Suitable for students in Grades 6-8, this problem reinforces understanding of linear segments and proportional relationships.

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