We are given a quadrilateral $ABCD$, where the following conditions hold:

• $AB = BC = BD$
• $\angle ABC = 90^\circ$

We are asked to find the length of $AC$.

Step-by-step solution:

1. Since $\angle ABC = 90^\circ$ and $AB = BC$, triangle $ABC$ is an isosceles right triangle. In such a triangle, the hypotenuse is $\sqrt{2}$ times the length of each leg.

2. In triangle $ABC$, we can use the Pythagorean theorem to find $AC$ (the hypotenuse): $AC = \sqrt{AB^2 + BC^2}$ Since $AB = BC$, we have: $AC = \sqrt{2 \times AB^2} = AB \sqrt{2}$ Therefore, $AC = AB \sqrt{2}$.

Since $AB = BC = BD$, the answer is: $AC = AB \sqrt{2}$

Would you like more details on this, or do you have further questions?

Here are 5 related questions to think about:

1. What is the relationship between the sides of an isosceles right triangle?
2. How would you approach this problem if the angle $\angle ABC$ was not $90^\circ$?
3. What would change if $AB \neq BC$?
4. Can you find the area of triangle $ABC$ given the information?
5. How does the length $BD$ impact other properties of quadrilateral $ABCD$?

Tip: The Pythagorean theorem is particularly useful for right triangles, especially when two sides are equal.