Question

the circumcenter of δabc was constructed as shown. which of the following statements is true about δabc and its circumcenter? options: lines k, m, and p represent the altitudes of δabc; the circumcenter is equidistant from each of the vertices of δabc; lines k, m, and p are perpendicular bisectors of δabc; lines k, m, and p are angle bisectors of δabc

Explanation:

Step1: Recall Circumcenter Properties

The circumcenter of a triangle is the intersection of the perpendicular bisectors of the triangle's sides. Perpendicular bisectors are lines that are perpendicular to a side and bisect it (divide it into two equal parts).

Step2: Analyze Each Option

• For the option about lines \( k \), \( m \), \( p \) being perpendicular bisectors: By the definition of circumcenter construction, the lines that intersect to form the circumcenter are the perpendicular bisectors of the triangle's sides. So we check the markings (right angles and bisected segments) in the diagram. The lines have right angles (perpendicular) and bisect the sides (marked with equal segments), so they are perpendicular bisectors.
• Other options: The circumcenter is equidistant from vertices (not sides, so that option is wrong). Lines being angle bisectors or something else doesn't match the circumcenter construction (angle bisectors relate to incenter, not circumcenter).

Answer:

The option stating "Lines \( k \), \( m \), and \( p \) are perpendicular bisectors of \( \triangle ABC \)" (exact wording depends on the option's text, but this is the correct property - the lines are perpendicular bisectors as they form the circumcenter by intersecting at the circumcenter, which requires perpendicular bisectors of sides).