Can a segment be both a median and a perpendicular bisector at the same time?

Yes, a segment can be both a median and a perpendicular bisector, but only under specific conditions. This happens in an **isosceles triangle** with a **vertex angle of 90 degrees**, which forms a **right isosceles triangle**.

Here's why:

1. A **median** is a segment from a vertex to the midpoint of the opposite side.
2. A **perpendicular bisector** is a segment that both bisects (divides into two equal parts) and is perpendicular to the segment it intersects.

In a right isosceles triangle, the altitude from the right angle to the hypotenuse serves both as the median and the perpendicular bisector of the hypotenuse. This is because:

- The hypotenuse is evenly divided by the altitude from the right angle vertex.
- The altitude is perpendicular to the hypotenuse due to the right angle at the vertex.

Thus, this altitude meets both conditions of being a median and a perpendicular bisector simultaneously.

Would you like more details on this concept or a visual example?

1. What other properties does a right isosceles triangle have?
2. In which other types of triangles might medians and perpendicular bisectors coincide?
3. How does this scenario differ in a scalene triangle?
4. What is the relationship between altitudes, medians, and perpendicular bisectors in an equilateral triangle?
5. Can an obtuse triangle have a median that is also a perpendicular bisector?

**Tip:** In geometry, when analyzing a special case, always consider symmetrical properties to determine unique segment relationships.

Discover how a segment can be both a median and a perpendicular bisector in specific conditions, like in a right isosceles triangle. This explanation covers essential geometry concepts, such as medians, perpendicular bisectors, and triangle properties. Ideal for students in Grades 8-10.

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