Question

in the diagram, ce=12 units and be=5 units. based on the given information, what is ae? 2 units 7 units 12 units 17 units

Explanation:

Step1: Identify perpendicular bisector

From the diagram, \( AB = BC \) (marked with ticks) and \( \angle ABE = 90^\circ \), so \( BE \) is the perpendicular bisector of \( AC \). Thus, \( AE = CE \)? Wait, no, wait—wait, actually, since \( BE \) is perpendicular bisector, \( AE = CE \)? Wait, no, wait, \( CE = 12 \), but wait, no, maybe triangle \( ABE \) and \( CBE \)? Wait, no, the markings: \( AB = BC \) (ticks), \( BE \perp AC \), so by SAS, triangles \( ABE \) and \( CBE \) are congruent. Therefore, \( AE = CE \)? Wait, no, \( CE = 12 \), but \( BE = 5 \). Wait, no, maybe I misread. Wait, the diagram: \( E \) is on the line, and \( CE = 12 \), \( BE = 5 \). Wait, no, actually, since \( BE \) is the perpendicular bisector (because \( AB = BC \) and right angle at \( B \)), then \( AE = CE \)? Wait, no, \( CE = 12 \), so \( AE = 12 \)? Wait, no, that can't be. Wait, maybe \( AE = CE \)? Wait, the options include 12. Wait, let's re-examine. The diagram has \( AB = BC \) (ticks), \( BE \perp AC \), so by the perpendicular bisector theorem, \( AE = CE \). Wait, \( CE = 12 \), so \( AE = 12 \)? But wait, \( BE = 5 \) is a distractor? Wait, the options: 2,7,12,17. So 12 is an option. Wait, maybe that's it.

Step2: Confirm congruence or theorem

Since \( BE \) is perpendicular to \( AC \) (right angle) and \( AB = BC \) (marked equal), triangle \( ABE \cong \triangle CBE \) by SAS ( \( AB = BC \), \( \angle ABE = \angle CBE = 90^\circ \), \( BE = BE \) ). Therefore, \( AE = CE \). Given \( CE = 12 \), so \( AE = 12 \).

Answer:

12 units (the option with 12 units)