The image shows a geometry problem involving two triangles, one inscribed within the other, and asks to find the lengths of segments $BC$ and $AC$ using similar triangles.

We are given the following information from the image:

• The right-angled triangle $ABC$ and another triangle $AXY$.
• Segment $AB = 5 \, \text{cm}$, segment $BX = 8 \, \text{cm}$, segment $AX = 6 \, \text{cm}$, and segment $XY = 10 \, \text{cm}$.

We will use the concept of similar triangles. The triangles $ABC$ and $AXY$ appear to be similar because they share an angle (at $A$) and both have right angles, implying angle-angle similarity.

Step 1: Set up the ratio of corresponding sides

For similar triangles, the ratios of corresponding sides are equal. So, we have:

$\frac{AB}{AX} = \frac{BC}{XY}$

Substitute the given values:

$\frac{5}{6} = \frac{BC}{10}$

Step 2: Solve for $BC$

We now solve for $BC$ by cross-multiplying:

$BC = \frac{5}{6} \times 10 = \frac{50}{6} = 8.33 \, \text{cm}$

Step 3: Solve for $AC$

Similarly, the ratio for the hypotenuses of the triangles is:

$\frac{AC}{AY} = \frac{AB}{AX}$

Since $AY = XY$ (they are the same length):

$\frac{AC}{10} = \frac{5}{6}$

Solving for $AC$:

$AC = \frac{5}{6} \times 10 = 8.33 \, \text{cm}$

Conclusion:

• The length of $BC$ is approximately $8.33 \, \text{cm}$.
• The length of $AC$ is also approximately $8.33 \, \text{cm}$.

Would you like further details or a different explanation for this solution?

Here are 5 questions to extend this problem:

1. How can we verify that the triangles are similar using the properties of angles?
2. What would happen if the ratio between two corresponding sides was different?
3. How can the Pythagorean theorem be used to verify the length of $BC$?
4. Can you calculate the area of both triangles?
5. What other methods could be used to find the unknown sides in similar triangles?

Tip: Always verify the similarity of triangles before applying the ratio of corresponding sides. This ensures the method is correct.