Question

6 match the correct answers.
given: ( \triangle abc ); ray ( overrightarrow{db} ) is perpendicular to ( overline{ac} ); ( ad = 8 ) in, ( overline{ab} cong overline{bc} )
what is the length of ( overline{cd} )?
by what theorem?

Explanation:

Step1: Use the property of isosceles triangles

In an isosceles triangle \( \triangle ABC\) with \( \overline{AB}\cong\overline{BC}\), if a line \( \overrightarrow{DB}\) is perpendicular to \( \overline{AC}\) (i.e., \( DB\perp AC\)), then by the Isosceles Triangle Theorem (which states that in an isosceles triangle, the altitude from the vertex angle to the base bisects the base).
Let \( AC\) be the base. Since \( AD = 8\) in and \( DB\) is the altitude (perpendicular from \( B\) to \( AC\)), and \( \triangle ABC\) is isosceles with \( AB = BC\).

Step2: Find the length of \( CD\)

By the Isosceles Triangle Theorem, \( AD=CD\). Given \( AD = 8\) in, so \( CD = 8\) in.

Answer:

The length of \( \overline{CD}\) is \( 8\) in. By the Isosceles Triangle Theorem.