Question

select the correct answer from each drop - down menu. in the figure, \\( \overline { a c } \\) and \\( \overline { b d } \\) bisect each other. the length of \\( \overline { a b } \\) is 12 centimeters and the length of \\( \overline { b c } \\) is 11 centimeters. the length of \\( \overline { a d } \\) is centimeters and the length of \\( \overline { d c } \\) is centimeters.

Explanation:

Step1: Prove triangles congruent

Since \(AC\) and \(BD\) bisect each other (\(AE = CE\), \(BE=DE\)), and \(\angle AEB=\angle CED\) (vertically - opposite angles), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle AEB\cong\triangle CED\). Similarly, \(\triangle AED\cong\triangle CEB\).

Step2: Use congruent - triangle properties

For congruent triangles \(\triangle AEB\) and \(\triangle CED\), \(AB = CD\). Given \(AB = 12\) cm, so \(CD=12\) cm.
For congruent triangles \(\triangle AED\) and \(\triangle CEB\), \(AD = BC\). Given \(BC = 11\) cm, so \(AD = 11\) cm.

Answer:

The length of \(\overline{AD}\) is \(11\) centimeters and the length of \(\overline{DC}\) is \(12\) centimeters.