1. Perpendicular Bisector of a Triangle
2. Circumcenter
3. Orthocenter
4. Concurrent
Math with the definitions
A. The common point of the perpendicular bisectors of a triangle
B. Segment from a vertex that is perpendicular to the opposite side or line containing the opp. side
C. Segment that is part of a perpendicular bisectors of one of the sides
D. The common point of the angle bisectors of a triangle
these definitions surely can be found in your textbook.
Start with 3A
1. Perpendicular Bisector of a Triangle:
- Step 1: Draw a triangle with three sides.
- Step 2: Find the midpoint of each side of the triangle.
- Step 3: Draw a line that passes through each midpoint and is perpendicular to the corresponding side. This line is called the perpendicular bisector.
- Step 4: Repeat steps 2 and 3 for the other two sides of the triangle.
- Step 5: The intersection point of the three perpendicular bisectors is the circumcenter of the triangle.
2. Circumcenter:
- Step 1: Draw a triangle with three sides.
- Step 2: Construct the perpendicular bisectors of each side using the steps mentioned in the "Perpendicular Bisector of a Triangle" explanation.
- Step 3: The point where the three perpendicular bisectors intersect is called the circumcenter.
3. Orthocenter:
- Step 1: Draw a triangle with three sides.
- Step 2: Locate the three altitudes of the triangle. An altitude is a line segment from a vertex that is perpendicular to the opposite side or the line containing the opposite side.
- Step 3: Extend each altitude so that they meet at a single point.
- Step 4: The point where the altitudes intersect is called the orthocenter.
4. Concurrent:
- Step 1: Concurrent means that three or more lines, segments, or rays intersect at a single point.
- Step 2: In the context of a triangle, the circumcenter is the intersection point of the perpendicular bisectors, and the orthocenter is the point where the altitudes meet.
- Step 3: The circumcenter and orthocenter of a triangle are often different points, but they can be concurrent if the triangle is acute or right-angled.
- Step 4: In an acute or right-angled triangle, the circumcenter, orthocenter, and the intersection point of the angle bisectors are concurrent at a single point called the incenter.
1. Perpendicular Bisector of a Triangle:
A perpendicular bisector of a triangle is a line or segment that divides a side of a triangle into two equal parts and is perpendicular to that side.
To find the perpendicular bisector of a triangle:
- Take any two points on one side of the triangle.
- Find the midpoint of the line segment connecting these two points.
- Construct a line that is perpendicular to the side of the triangle at the midpoint.
- Repeat this process for the other two sides of the triangle.
- In the end, the three perpendicular bisectors will intersect at a single point called the circumcenter.
2. Circumcenter:
The circumcenter of a triangle is the point of intersection of the perpendicular bisectors of the triangle's sides. It is equidistant from all three vertices of the triangle.
To find the circumcenter of a triangle:
- Follow the steps to find the perpendicular bisectors of the triangle's sides.
- Where the three perpendicular bisectors intersect is the circumcenter of the triangle.
3. Orthocenter:
The orthocenter of a triangle is the point of intersection of the lines containing the altitudes of the triangle. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side.
To find the orthocenter of a triangle:
- Draw the altitudes of the triangle, which are lines drawn from each vertex of the triangle perpendicular to the respective opposite side.
- The point where the three altitudes intersect is the orthocenter of the triangle.
4. Concurrent:
In geometry, the term "concurrent" refers to lines, segments, or rays that intersect at the same point.
In the context of the given definitions:
- The perpendicular bisectors of a triangle are concurrent at the circumcenter (option A).
- The altitudes of a triangle are concurrent at the orthocenter (option B).
- The perpendicular bisectors of one of the sides of a triangle are concurrent (option C).
- The angle bisectors of a triangle are concurrent at the incenter (option D), which was not mentioned in the original definitions.