To find the lengths of $AD$ and $BD$, we'll use the property of similar triangles, which we established earlier:

• Given: $\triangle ABC \sim \triangle ADC$ (from the previous problem).
• Known lengths:
  • $AB = 16 \, \text{cm}$
  • $BC = 20 \, \text{cm}$
  • $AC = 24 \, \text{cm}$

Since the triangles $\triangle ABC$ and $\triangle ADC$ are similar, their corresponding sides are proportional:

$\frac{AB}{AD} = \frac{BC}{CD} = \frac{AC}{AC}$

However, since the lengths $BC$ and $AC$ are on different triangles, we'll actually use the similar triangles condition with the segments:

$\frac{AB}{AD} = \frac{BC}{BD} = \frac{AC}{CD}$

Step 1: Express $BD$ and $CD$

Let's denote $BD = x$ and $CD = y$.

From $\frac{AB}{AD} = \frac{BC}{BD}$:

$\frac{16}{AD} = \frac{20}{x}$

Similarly, from $\frac{AB}{AD} = \frac{AC}{CD}$:

$\frac{16}{AD} = \frac{24}{y}$

But we know $AD = AB + BD$, and $AB = 16$ cm.

Step 2: Solve the system of equations

1. $\frac{16}{AD} = \frac{20}{x}$

   $AD = \frac{16x}{20} = \frac{4x}{5}$

2. $\frac{16}{AD} = \frac{24}{y}$

   $AD = \frac{16y}{24} = \frac{2y}{3}$

Since $AD = AB + BD$, substitute:

$\frac{4x}{5} = 16 + x$

Solving for $x$ (i.e., $BD$):

$4x = 80 + 5x$

$x = 80$

So, $BD = 80$ cm.

Similarly, solve for $y$ using the second equation:

$\frac{2y}{3} = 24 + y$

$2y = 72 + 3y$

$y = 72$

Thus, $CD = 72$ cm.

Since the two triangles are similar, and we assumed the triangle sides might be overlapping or continuing, you should check with your diagram to ensure that the results match the physical lengths.

But if $BD$ and $CD$ are considered external, and $AD$ is shorter:

Given that $AB = 16$, $AD = \text{same as BD using similar proportions} = \frac{20}{16}*24=30$.

Then you would use the simple side lengths for final proportions.

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Would you like more details or have any questions?

Here are some related questions to expand on this concept:

1. How do we apply the Pythagorean theorem in these cases?
2. How can we use trigonometric identities to find missing side lengths?
3. How do we apply properties of right triangles to find $AD$ and $BD$?
4. What are other methods to determine segment lengths in similar triangles?
5. Can $AD$ and $BD$ be determined using coordinate geometry?
6. How does the concept of ratios apply in these triangle segments?
7. What are some real-life applications of similar triangles?
8. Can $AD$ and $BD$ be found using vector algebra?

Tip: Always verify with similar triangle properties that all proportions match when solving for unknown lengths in complex geometric figures.