Perpendicular Bisectors and Circumcircle
🎬 Video: Perpendicular Bisector Construction and Use
What is a Perpendicular Bisector? (0:01)
A perpendicular bisector is a line that cuts a line segment into two equal parts and meets it at a 90-degree angle.
This means it always passes through the midpoint of the line segment while being perpendicular to it.
Perpendicular bisectors are useful in geometry for finding equidistant points and constructing triangles.
How to Draw a Perpendicular Bisector? (0:34)
There are two common methods to draw a perpendicular bisector:
Method 1: Using a Ruler
- Find the midpoint of the given line segment. For example, if the segment is 6 cm long, the midpoint is at 3 cm.
- Draw a perpendicular line through the midpoint using a right-angle ruler. This ensures the line forms a 90-degree angle with the original segment.
Method 2: Using a Compass
- Place the compass at one endpoint of the segment and draw an arc above and below the line.
- Repeat from the other endpoint using the same radius to create two intersection points.
- Draw a straight line through these intersection points—this is the perpendicular bisector.
Key Property of the Perpendicular Bisector (1:23)
A crucial property of a perpendicular bisector is that every point on the perpendicular bisector is equidistant from the segment’s endpoints.
How to Construct the Circumcenter of a Triangle? (1:35)
Key Terms in Circumcenter Construction:
- Circumcenter: The point where the perpendicular bisectors of all three sides of a triangle meet.
- Circumradius: The distance from the circumcenter to any vertex of the triangle.
- Circumcircle: The circle centered at the circumcenter that passes through all three vertices of the triangle.
Steps to Construct the Circumcircle:
- Draw the Perpendicular Bisectors and Locate the Circumcenter
- For each side of the triangle, locate its midpoint.
- Draw a perpendicular bisector through each midpoint, ensuring it forms a 90-degree angle with the side.
- The circumcenter is the point where all three perpendicular bisectors intersect.
- Find the Circumradius
- The circumradius is the distance from the circumcenter to any vertex of the triangle.
- You measure this distance using a ruler or compass
- Draw the Circumcircle
- Place the compass at the circumcenter and adjust it to the circumradius.
- Draw a circle that passes through all three vertices. This is the circumcircle of the triangle.
📂 Flashcards: How to Draw Perpendicular Bisector and Circumcircle of a Triangle
🍪 Quiz: Test Your Skills with Perpendicular Bisector and Circumcenter of a Triangle
🎩 Struggling with Geometry Concepts? Try AI Math Solver
Need math help? Chat with our AI Math Solver at the bottom right — available 24/7 for instant answers.