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Bisectors in a Triangle

Learn how to construct and use perpendicular bisectors in triangles to find the circumcenter, along with related properties and problem-solving techniques.

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Bisectors in a Triangle

Study Guide

Key Definition

A bisector in a triangle is a line that cuts an angle or side into two equal parts. The circumcenter is the point where the perpendicular bisectors of a triangle intersect.

Important Notes

The circumcenter is equidistant from the vertices of the triangle.
The circumcenter lies inside the triangle for acute triangles, on the hypotenuse for right triangles and outside the triangle for obtuse triangles.

Mathematical Notation

$\angle$ represents an angle.

$\triangle$ represents a triangle.

$\overline{AB}$ represents the line segment AB.

Remember to use the proper notation when solving problems.

Why It Works

The circumcenter is equidistant from the vertices of the triangle, making it the center of the smallest circle in which the triangle can be inscribed.

Remember

The circumcenter is the point of concurrency of the perpendicular bisectors of a triangle.

Quick Reference

Circumcenter:$O$

Understanding Bisectors in a Triangle

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A perpendicular bisector of a side of a triangle is a line that divides the side into two equal parts and is perpendicular to it. In any triangle, the three perpendicular bisectors intersect at a single point called the circumcenter, which is equidistant from all vertices.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Where does the circumcenter of an obtuse triangle lie?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Natha, Hiren and Joe's homes represent three non-collinear points on a coordinate plane. If they want to meet at a common place such that each one will have to travel the same distance from their homes, how will you decide the meeting point?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $\triangle ABC$ is an isosceles right triangle, where does the circumcenter lie?

Challenge Quiz

Single Choice Quiz

Advanced

If $\triangle ABC$ is such that $AB = BC$ and $\angle ABC = 90\degree$, what is the circumcenter?

Recap

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Review key concepts and takeaways