Question

1. match.

given: \\( \triangle a b c ; \overleftrightarrow { b d } \perp \overline { a c } ; \overline { a d } \cong \overline { d c } ; b c = 7 \\) inches
what is the length of \\( \overline { a b } \\)?
by which theorem?

Explanation:

Step1: Recall the Perpendicular Bisector Theorem

The Perpendicular Bisector Theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
In \(\triangle ABC\), \(\overleftrightarrow{BD}\perp\overline{AC}\) and \(\overline{AD}\cong\overline{DC}\). So, \(BD\) is the perpendicular bisector of \(AC\).

Step2: Apply the theorem

Since \(B\) is a point on the perpendicular bisector of \(AC\), by the Perpendicular Bisector Theorem, \(AB = BC\).
Given \(BC = 7\) inches.

Answer:

The length of \(\overline{AB}\) is \(7\) inches. By the Perpendicular Bisector Theorem.