Question

point z is the circumcenter of \\( \triangle lmn \\). which must be true? \\( \overline { a n } \cong \overline { l b } \\) \\( \overline { n c } \cong \overline { b z } \\) \\( \angle b m z \cong \angle c m z \\) \\( \angle z b m \cong \angle z c m \\)

Explanation:

Step1: Recall the property of circumcenter

The circumcenter of a triangle is equidistant from the vertices of the triangle. So, \(ZM = ZN=ZL\). Also, since \(ZB\perp LM\) and \(ZC\perp MN\) (perpendicular bisectors), in right - triangles \( \triangle ZBM\) and \( \triangle ZCM\):

• \(ZM = ZM\) (common side)
• \(ZB = ZC\) (distance from circumcenter to the sides as the perpendicular bisectors are equal in length for a circumcenter)

Step2: Use the Hypotenuse - Leg (HL) congruence criterion

For right - triangles \( \triangle ZBM\) and \( \triangle ZCM\), by the HL (Hypotenuse - Leg) congruence criterion (\(ZM\) is the hypotenuse and \(ZB = ZC\) as legs), \( \triangle ZBM\cong\triangle ZCM\).

Step3: Use the property of congruent triangles

If \( \triangle ZBM\cong\triangle ZCM\), then their corresponding angles are congruent. So, \(\angle ZBM\cong\angle ZCM\)

Answer:

\(\angle ZBM\cong\angle ZCM\)