https://file.aiquickdraw.com/mathbot/file/1726971141581w3mb0x1i.jpg

From the image, we can analyze the given isosceles triangle \( ABC \) with the following provided information:

- \( AB = 8 \) cm
- \( AD = 17 \) cm
- We are asked to find \( BD \) and \( AC \), two unknown lengths in the diagram.

### Analysis:

1. **Isosceles triangle property**: Since triangle \( ABC \) is isosceles, sides \( AB = AC \).
2. **Right angle**: The triangle has a right angle at \( B \), implying that triangle \( ABD \) can be divided into right triangles. This might allow us to use the Pythagorean theorem.
3. **Line \( BD \)**: Since there is a right angle at \( B \), we can use the Pythagorean theorem in triangle \( ABD \) to calculate \( BD \).

We will start by solving for \( BD \) using the Pythagorean theorem. Given \( AB = 8 \) cm and \( AD = 17 \) cm, the formula will be:

\[
AD^2 = AB^2 + BD^2
\]

Let me compute the length of \( BD \) first.Using the Pythagorean theorem, we find that the length of \( BD \) is 15 cm.

Next, since triangle \( ABC \) is isosceles, \( AC \) will be equal to \( AB \). Therefore, the length of side \( AC \) is:

\[
AC = 8 \, \text{cm}
\]

### Final answers:
- \( BD = 15 \, \text{cm} \)
- \( AC = 8 \, \text{cm} \)

Do you want more details or have any questions?

Here are five related questions to further explore:
1. What is the perimeter of triangle \( ABD \)?
2. Can you calculate the area of triangle \( ABC \)?
3. What would happen if \( AB \) and \( AC \) were unequal? How would that affect the triangle?
4. How does the length \( AD \) affect the shape of the triangle?
5. Can we apply the cosine rule to calculate any angle in this scenario?

**Tip**: Always double-check for symmetry in isosceles triangles, as it simplifies many calculations!

Triangle ABC is an isosceles triangle. Side AB is 8 cm. Side AD is 17 cm. Find side BD and side AC.

In this problem, we solve for the lengths of side BD and side AC in an isosceles triangle using the Pythagorean theorem. Given side AB = 8 cm and side AD = 17 cm, learn how to apply the formula to calculate the unknown lengths.

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