Question

point z is the circumcenter of δtuv. what is m∠zub? 23.7° 32.8° 33.5° 57.2°

Explanation:

Step1: Recall circumcenter properties

The circumcenter \( Z \) of a triangle is equidistant from the vertices, and the perpendicular bisectors of the sides meet at \( Z \). So, \( ZB \perp TU \) (since \( B \) is on the perpendicular bisector of \( TU \)), meaning \( \triangle ZUB \) is a right triangle with \( \angle ZBU = 90^\circ \).

Step2: Find \( \angle ZUB \)

In \( \triangle ZUT \), we know \( \angle UTZ = 32.8^\circ \), and \( Z \) is the circumcenter, so \( ZT = ZU \) (radii of the circumcircle). Wait, alternatively, in right triangle \( ZUB \), we know one angle in the larger context. Wait, the angle at \( Z \) between \( ZU \) and \( ZT \) – no, let's look at the right angle at \( B \). So in \( \triangle ZUB \), \( \angle ZBU = 90^\circ \), and we can find \( \angle ZUB \) by knowing that the sum of angles in a triangle is \( 180^\circ \). Wait, the angle \( \angle TZU \) is \( 57.2^\circ \)? No, wait, the diagram shows \( \angle TZU = 57.2^\circ \)? Wait, no, the angle at \( Z \) between \( ZT \) and \( ZU \) – actually, since \( ZB \) is perpendicular to \( TU \), and \( ZA \) is perpendicular to \( TV \), etc. Wait, another approach: in a right triangle, the two acute angles are complementary. Wait, we know that \( \angle UTZ = 32.8^\circ \), and since \( Z \) is the circumcenter, \( ZT = ZU \), so \( \triangle ZTU \) is isoceles? No, wait, the key is that \( \angle ZUB \) and \( \angle UTZ \) – wait, no. Wait, the right angle at \( B \), so \( \angle ZUB + \angle TZU = 90^\circ \)? Wait, no, let's calculate. The sum of angles in \( \triangle ZUB \): \( \angle ZBU = 90^\circ \), so \( \angle ZUB + \angle BZU = 90^\circ \). Wait, but we know that \( \angle TZU = 57.2^\circ \)? Wait, no, the diagram has \( \angle TZU = 57.2^\circ \)? Wait, the angle marked \( 57.2^\circ \) is \( \angle ZUT \)? No, the angle at \( Z \) between \( ZU \) and \( ZT \) is \( 57.2^\circ \)? Wait, maybe I made a mistake. Wait, the correct approach: since \( ZB \) is perpendicular to \( TU \), \( \angle ZBU = 90^\circ \). Then, in \( \triangle ZUB \), \( \angle ZUB = 90^\circ - 32.8^\circ = 57.2^\circ \)? No, that's not right. Wait, no, the angle \( \angle UTZ = 32.8^\circ \), and since \( Z \) is the circumcenter, \( ZT = ZU \), so \( \angle ZUT = \angle UTZ = 32.8^\circ \)? No, that's not. Wait, maybe the angle \( \angle ZUB \) is complementary to \( 57.2^\circ \)? Wait, \( 90^\circ - 57.2^\circ = 32.8^\circ \). Ah! Yes, because in the right triangle \( ZUB \), \( \angle ZBU = 90^\circ \), so \( \angle ZUB + \angle BZU = 90^\circ \). If \( \angle BZU = 57.2^\circ \), then \( \angle ZUB = 90^\circ - 57.2^\circ = 32.8^\circ \). Wait, that makes sense. So \( 90 - 57.2 = 32.8 \). So \( m\angle ZUB = 32.8^\circ \).

Answer:

\( 32.8^\circ \) (corresponding to the option B. \( 32.8^\circ \))