Question

select all statements that are true about equilateral triangle abc.
$x - 6\sqrt{3}$
$x = 3\sqrt{3}$
bd and cd are both 3 units long.
angles b and c are 60 degrees.
triangle abd is congruent to triangle acd.

Explanation:

Step1: Analyze the properties of an equilateral triangle

In an equilateral triangle, all angles are \(60^{\circ}\), so angles \(B\) and \(C\) are \(60\) degrees.

Step2: Use the properties of a \(30 - 60-90\) triangle

Since \(AD\) is the altitude of equilateral triangle \(ABC\), \(\triangle ACD\) is a \(30 - 60-90\) triangle. In \(\triangle ACD\), \(AC = 6\) (side of equilateral triangle), \(\angle C=60^{\circ}\), \(\angle CAD = 30^{\circ}\). The side opposite \(30^{\circ}\) (\(CD\)) is half of the hypotenuse (\(AC\)), so \(CD=\frac{AC}{2}=3\). Using the Pythagorean theorem in \(\triangle ACD\): \(x=\sqrt{AC^{2}-CD^{2}}=\sqrt{6^{2}-3^{2}}=\sqrt{36 - 9}=\sqrt{27}=3\sqrt{3}\).

Step3: Check congruence

In \(\triangle ABD\) and \(\triangle ACD\), \(AB = AC\) (equilateral triangle), \(AD=AD\) (common side), \(BD = CD\) (proven \(CD = 3\) and \(BD=BC - CD\), \(BC=6\) so \(BD = 3\)). By SSS (Side - Side - Side) congruence criterion, \(\triangle ABD\cong\triangle ACD\).

Answer:

\(x = 3\sqrt{3}\), \(BD\) and \(CD\) are both \(3\) units long, Angles \(B\) and \(C\) are \(60\) degrees, Triangle \(ABD\) is congruent to triangle \(ACD\)