Question

find the values of x and y.
x = □ (simplify your answer.)

Explanation:

Step1: Use the property of the median in a triangle

Since \(D\) is the mid - point of \(AC\) (\(AD = DC\)) and \(BD\) is a median. In \(\triangle ABC\), if \(BD\) is a median and \(AB = BC\) (from the markings), then \(\triangle ABD\cong\triangle CBD\) (by SSS congruence criterion: \(AD = CD\), \(AB = CB\), \(BD=BD\)).

Step2: Use the angle - side relationship

In \(\triangle ABD\), \(\angle A = 49^{\circ}\). Since \(\triangle ABD\cong\triangle CBD\), \(\angle ABD=\angle CBD\). Also, \(AB = BC\), so \(\angle A=\angle BCA = 49^{\circ}\).
In \(\triangle ABC\), \(\angle ABC=180^{\circ}-2\times49^{\circ}=82^{\circ}\). Then \(x = \frac{1}{2}\angle ABC\) (because \(BD\) is the angle - bisector as \(\triangle ABD\cong\triangle CBD\)), so \(x = 41^{\circ}\).

Step3: Use the mid - point property for length

Since \(D\) is the mid - point of \(AC\) and \(AC = 5\), then \(y=\frac{5}{2}=2.5\)

Answer:

\(x = 41\), \(y = 2.5\)