Question

Find the length of CD
Notice BD and AC are perpendicular (90 degrees) and AB=BC. That means BD is the perpendicular bisector of AC!



Now we know any point (like D) on the perpendicular bisector is the same distance away from the endpoints (A and C).



Set `8x-4` equal to `3x+1` and solve. Then plug it back in to find CD

Answer 1

Let's start by setting the expressions for the distances equal to each other, as you mentioned. First, we have the following equations for the distances from point D to points A and C:

- The distance from D to A: \( 8x - 4 \)
- The distance from D to C: \( 3x + 1 \)

Since BD is the perpendicular bisector of AC, these distances must be equal. Therefore, we can set up the equation:

\[
8x - 4 = 3x + 1
\]

Next, we will solve for \( x \):

1. Subtract \( 3x \) from both sides:
\[
8x - 3x - 4 = 1
\]
which simplifies to:
\[
5x - 4 = 1
\]

2. Next, add \( 4 \) to both sides:
\[
5x = 1 + 4
\]
which simplifies to:
\[
5x = 5
\]

3. Now, divide both sides by \( 5 \):
\[
x = 1
\]

Now that we have \( x \), we can substitute it back into one of the original distance equations to find the length of \( CD \). We can use either expression, but let's use \( 3x + 1 \):

\[
CD = 3(1) + 1 = 3 + 1 = 4
\]

Thus, the length of \( CD \) is:

\[
\boxed{4}
\]