Question

1. match.

given: \\( \triangle a b c ; \overleftrightarrow { b d } \perp \overline { a c } ; a b = b c ; a d = 5 \\) inches
what is the length of \\( \overline { d c } \\) ?
by which theorem?

Explanation:

Step1: Recall the property of isosceles triangles

In an isosceles triangle \( \triangle ABC\) with \(AB = BC\), and \(BD\perp AC\) (the altitude from the vertex \(B\) to the base \(AC\)).

Step2: Apply the Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that in an isosceles triangle, the altitude from the vertex angle to the base also bisects the base. So, \(AD=DC\).
Since \(AD = 5\) inches, then \(DC=5\) inches.

Answer:

The length of \(\overline{DC}\) is \(5\) inches. By the Isosceles Triangle Theorem.