Question

$\overleftrightarrow{rt}$ is a perpendicular bisector of $\overline{uv}$.
what is true of any triangle created by points u, v, and any point on $\overleftrightarrow{rt}$ other than s?
it will be an isosceles triangle.
it will be an equilateral triangle.
it will be a right triangle.
it will be an acute triangle.

Explanation:

Step1: Recall Perpendicular Bisector Property

A perpendicular bisector of a segment means any point on it is equidistant from the endpoints of the segment. Here, \( \overleftrightarrow{RT} \) is the perpendicular bisector of \( \overline{UV} \), so any point (let's say \( P \)) on \( \overleftrightarrow{RT} \) (other than \( S \)) will have \( PU = PV \).

Step2: Analyze Triangle Type

In triangle \( \triangle PUV \), if two sides (\( PU \) and \( PV \)) are equal, by the definition of an isosceles triangle (a triangle with at least two equal sides), \( \triangle PUV \) is isosceles. It may not be right (unless angles are \( 90^\circ \), not guaranteed), equilateral (needs all sides equal, not guaranteed), or acute (angles could be obtuse, e.g., if \( P \) is far from \( S \), angle at \( P \) could be obtuse).

Answer:

It will be an isosceles triangle.