Question

1. examine the given figure. the circumcenter of $\triangle abc$ was constructed as shown.

which of the following statements is true about $\triangle abc$ and its circumcenter?
\bigcirc lines k, m, and p are perpendicular bisectors of $\triangle abc$
\bigcirc lines k, m, and p are angle bisectors of $\triangle abc$
\bigcirc the circumcenter is equidistant from each of the sides of $\triangle abc$
\bigcirc lines k, m, and p represent the altitudes of $\triangle abc$

Explanation:

To determine the correct statement about the circumcenter of \(\triangle ABC\), we recall the properties of the circumcenter:

1. The circumcenter is the intersection of the perpendicular bisectors of the triangle's sides.
2. Let's analyze each option:

• Option 1: Lines \(k\), \(m\), and \(p\) are perpendicular bisectors of \(\triangle ABC\)

The circumcenter is formed by the intersection of the perpendicular bisectors of the triangle's sides. In the diagram, lines \(k\), \(m\), and \(p\) intersect at the circumcenter, and their markings (right angles, bisecting segments) suggest they are perpendicular bisectors.

• Option 2: Lines \(k\), \(m\), and \(p\) are angle bisectors

Angle bisectors split angles into two equal parts, but the circumcenter is not related to angle bisectors (that's the incenter). So this is incorrect.

• Option 3: The circumcenter is equidistant from each of the sides

The incenter (not circumcenter) is equidistant from the sides. The circumcenter is equidistant from the vertices. So this is incorrect.

• Option 4: Lines \(k\), \(m\), and \(p\) represent the altitudes

Altitudes are perpendicular to sides but do not necessarily bisect them. The circumcenter requires bisecting the sides, so this is incorrect.

Answer:

Lines \(k\), \(m\), and \(p\) are perpendicular bisectors of \(\triangle ABC\)