Question

proving the 30°-60°-90° triangle side length relationship
consider that △abc is an equilateral triangle, and \\(\overline{ad}\\) is a perpendicular bisector of △abc. if ab = 2x, complete the statements below.
\\(\square^2 + (ad)^2 = (2x)^2\\)
\\(\square^2 = \square x^2 - x^2\\)
\\(\square (ad)^2 = \square x^2\\)
\\(\square ad = \square \sqrt{3}\\)

Explanation:

Step1: Identify BD length

Since $\overline{AD}$ is the perpendicular bisector of equilateral $\triangle ABC$, $BD = \frac{1}{2}AB$. Given $AB=2x$, so $BD = x$.
Pythagorean theorem for $\triangle ABD$: $BD^2 + (AD)^2 = (2x)^2$
Substitute $BD=x$: $x^2 + (AD)^2 = (2x)^2$

Step2: Rearrange to solve for $AD^2$

Isolate $(AD)^2$: $(AD)^2 = (2x)^2 - x^2$
Calculate $(2x)^2$: $(AD)^2 = 4x^2 - x^2$

Step3: Simplify the expression

Combine like terms: $(AD)^2 = 3x^2$

Step4: Solve for $AD$

Take square root of both sides: $AD = \sqrt{3x^2} = x\sqrt{3}$

Answer:

1. $x$
2. $4$
3. $3$
4. $x$