Question

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

Explanation:

Step1: Apply Pythagorean theorem

In right - triangle \(ABD\), by Pythagorean theorem \(BD^{2}+(AD)^{2}=AB^{2}\). Since \(\triangle ABC\) is equilateral and \(AD\) is a perpendicular bisector, \(BD=\frac{AB}{2}\). Given \(AB = 2x\), then \(BD=x\). So \(x^{2}+(AD)^{2}=(2x)^{2}\).

Step2: Simplify the equation for \((AD)^{2}\)

Expand \((2x)^{2}=4x^{2}\). Then \((AD)^{2}=(2x)^{2}-x^{2}=4x^{2}-x^{2}\).

Step3: Further simplify \((AD)^{2}\)

\((AD)^{2}=3x^{2}\).

Step4: Solve for \(AD\)

Take the square root of both sides. \(AD=\sqrt{3x^{2}}=x\sqrt{3}\) (since \(AD>0\)).

Answer:

\(x\); \(4\); \(3\); \(x\)