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? \bigcirc it will be a right triangle. \bigcirc it will be an acute triangle. \bigcirc it will be an equilateral triangle. \bigcirc it will be an isosceles triangle.

Explanation:

Since \( \overleftrightarrow{RT} \) is the perpendicular bisector of \( \overline{UV} \), any point on \( \overleftrightarrow{RT} \) (other than \( S \)) is equidistant from \( U \) and \( V \) (by the Perpendicular Bisector Theorem). In a triangle with vertices \( U \), \( V \), and a point \( P \) on \( \overleftrightarrow{RT} \), \( PU = PV \), so the triangle has two equal - length sides. By the definition of an isosceles triangle (a triangle with at least two equal - length sides), the triangle \( \triangle UPV \) is isosceles. It is not necessarily a right triangle (unless specific angle conditions are met), not necessarily acute (could be obtuse depending on the position of \( P \)), and not equilateral (since \( UV \) and \( PU \) don't have to be equal).

Answer:

D. It will be an isosceles triangle.