Let’s walk through the missing steps in Naomi’s solution.

Given:

• $\overleftrightarrow{BD}$ is the perpendicular bisector of segment $\overline{AC}$.
• $\overline{AC}$ is 8 units long, meaning it’s split equally by the bisector.
• $\overline{BD}$ is 3 units long.

Fill-in-the-blank solution:

1. $\overleftrightarrow{BD}$ bisects $\overline{AC}$, so both $\overline{AB}$ and $\overline{BC}$ are
   4 units long (half of 8).

2. We are given that $\overline{BD}$ is 3 units long.

3. $\overleftrightarrow{BD}$ is perpendicular to $\overline{AC}$, so $\triangle ABD$ is a right triangle.

4. Applying the Pythagorean theorem to $\triangle ABD$:

   AB^2 + BD^2 = AD^2 \quad \text{(hypotenuse is AD)}

   Substituting:

   $4^2 + 3^2 = AD^2$ $16 + 9 = AD^2$ $AD^2 = 25 \quad \Rightarrow \quad AD = \sqrt{25} = 5$

   So, the hypotenuse $AD$ is 5 units long.

5. A similar process can be applied to $\triangle CBD$, where:

   • $BC = 4$ units, $BD = 3$ units.
   • By the Pythagorean theorem, $CD = 5$ units.

Thus, both $AD$ and $CD$ are 5 units long, which means that point $D$ is equidistant from points $A$ and $C$.

Final Answer:

• $\overline{AB}$ and $\overline{BC}$ are 4 units long.
• The hypotenuse is 5 units long.
• Both distances $AD$ and $CD$ are equal, so point $D$ is equidistant from $A$ and $C$.

Do you need further clarification or help with the geometry concepts?

5 Related Questions:

1. How does the Pythagorean theorem apply to other right triangles?
2. What are the properties of a perpendicular bisector in any triangle?
3. Can two triangles with the same side lengths be different in shape?
4. How would the result change if $\overline{AC}$ were a different length?
5. What is the relationship between a bisector and the symmetry of a figure?

Tip: Always identify the right triangle's hypotenuse and apply the Pythagorean theorem carefully.