In the figure, square WXYZ has a diagonal of 12 units. Point A is a midpoint of segment WX, segment AB is perpendicular to segment AC and AB = AC. What is the length of segment BC?

Don't worry your only 3 years late AoPS

To find the length of segment BC, we can use the Pythagorean theorem.

First, let's visualize the given information. Square WXYZ has a diagonal that connects points W and Z, which measures 12 units. Point A is the midpoint of segment WX. Segment AB is perpendicular to segment AC, and AB is equal in length to AC.

Let's consider triangle ABC. Segment AC represents the hypotenuse, and segment AB and BC are the legs of the right triangle.

We know that AB = AC. Let's denote this length as x. So, AB = x and AC = x.

We also know that segment AB is perpendicular to segment AC, which means that angles B and C are right angles.

Now, let's use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

In triangle ABC, we have:

AB^2 + BC^2 = AC^2

Since AB = AC = x, we can substitute and get:

x^2 + BC^2 = x^2

Simplifying the equation, we have:

BC^2 = x^2 - x^2
BC^2 = 0
BC = 0

Therefore, the length of segment BC is 0 units.

It's important to note that segment BC doesn't exist in this scenario, as the triangle ABC degenerates into a straight line.

no idea. Where are B and C?

Nice job cheating Jiskha!