Question

which type of triangle will always have a perpendicular bisector that is also an angle bisector?
equilateral
right
scalene
obtuse

Explanation:

Step1: Recall triangle properties

In an equilateral triangle, all sides are equal, and all angles are \(60^\circ\). The perpendicular bisector of a side (which is also a median and an angle bisector) splits the angle into two equal \(30^\circ\) angles? Wait, no, in an equilateral triangle, the perpendicular bisector of a side: since all sides are equal, the perpendicular bisector from a vertex to the opposite side will also bisect the angle (because the triangle is symmetric in all aspects). For a right triangle, the perpendicular bisector of the hypotenuse is not an angle bisector (unless it's an isosceles right triangle, but right triangles in general aren't). A scalene triangle has all sides and angles different, so the perpendicular bisector of a side won't bisect the angle (since the angles are different, the symmetry isn't there). An obtuse triangle (non - isosceles) also won't have the perpendicular bisector as an angle bisector. But in an equilateral triangle, due to its symmetry (all sides equal, all angles equal), the perpendicular bisector of any side is also the angle bisector of the corresponding angle.

Step2: Analyze each option

• Equilateral: As explained, in an equilateral triangle, the perpendicular bisector of a side is also the angle bisector of the angle at the vertex from which the perpendicular is drawn, because of the equal sides and angles.
• Right: In a right triangle (e.g., a \(3 - 4 - 5\) triangle), the perpendicular bisector of the hypotenuse is not an angle bisector of the right angle (the right angle is \(90^\circ\), and the perpendicular bisector of the hypotenuse doesn't split it into two equal angles in a non - isosceles right triangle).
• Scalene: A scalene triangle has all sides of different lengths and all angles of different measures. So the perpendicular bisector of a side will not bisect the angle (since the angles adjacent to the side are different, the line that bisects the side at right angles won't bisect the angle).
• Obtuse: An obtuse triangle (non - isosceles) has one angle greater than \(90^\circ\). The perpendicular bisector of a side will not bisect the angle (due to the unequal sides and angles).

Answer:

equilateral