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!