Let's solve the given problem step by step. The triangle $\triangle ABC$ is an equilateral triangle with side length $20$, and $D$ is the midpoint of $\overline{AC}$, forming two 30-60-90 triangles.

Step 1: Understand the properties of a 30-60-90 triangle

In a 30-60-90 triangle:

• The ratio of the sides is $1 : \sqrt{3} : 2$, where:
  • The shortest side (opposite the 30° angle) is half the hypotenuse.
  • The side opposite the 60° angle is $\sqrt{3}$ times the shortest side.

Step 2: Break down the triangle

Since $\triangle ABC$ is equilateral:

• $\overline{AB} = \overline{BC} = \overline{AC} = 20$.
• $D$ is the midpoint of $\overline{AC}$, so $\overline{AD} = \overline{DC} = 10$.

Step 3: Solve for $\overline{BD}$

In $\triangle BDC$:

• $\overline{BC} = 20$ (hypotenuse).
• $\overline{DC} = 10$ (short side).
• $\overline{BD}$ is the side opposite the 60° angle.

From the 30-60-90 triangle ratio: $\overline{BD} = \overline{DC} \cdot \sqrt{3} = 10 \cdot \sqrt{3}.$

Final Answer:

The length of $\overline{BD}$ is: $\boxed{10\sqrt{3}}$

Let me know if you want further clarification or have any related questions!

Related Questions:

1. What is the perimeter of triangle $\triangle ABC$?
2. How can you calculate the area of triangle $\triangle ABC$?
3. What are the properties of an equilateral triangle?
4. How do 30-60-90 triangle ratios apply in real-world problems?
5. How would the solution change if $\overline{AB} = 24$?

Tip:

Always memorize the side ratios for special right triangles like 30-60-90 and 45-45-90—they simplify many geometric problems!