Question

given: ( p ) is the circumcenter of ( \triangle abc ); ( overline{dp},overline{ep} ), and ( overline{fp} ) are perpendicular bisectors; ( ap = 25mathrm{mm} ).
what is the length of ( overline{bp} )?
what is the length of ( overline{cp} )?

1. choose the correct answer.

the ____ is a segment that extends from the vertex of a triangle to the opposite side and is perpendicular to the side of the triangle.
altitude
incenter
median

Explanation:

Step1: Recall the property of the circum - center

The circum - center of a triangle is equidistant from the vertices of the triangle.

Step2: Apply the property to find \(BP\) and \(CP\)

Since \(P\) is the circum - center of \(\triangle ABC\), \(AP = BP=CP\). Given \(AP = 25\space mm\), so \(BP = 25\space mm\) and \(CP = 25\space mm\).

Step3: Define the terms for the multiple - choice question

• An altitude is a segment that extends from the vertex of a triangle to the opposite side and is perpendicular to the side of the triangle.
• The in - center is the center of the inscribed circle of a triangle (the point of intersection of the angle bisectors).
• A median is a segment that extends from the vertex of a triangle to the mid - point of the opposite side.

Answer:

The length of \(\overline{BP}\) is \(25\space mm\).
The length of \(\overline{CP}\) is \(25\space mm\).
For the multiple - choice question: A. altitude.