Note: Your teacher will grade your response to questions 13–15 to ensure you receive proper credit for your answer. For a triangle, list the respective names of the points of concurrency of perpendicular bisectors of the sides bisectors of the angles medians lines containing the altitudes (4 points)
The respective names of the points of concurrency for a triangle are as follows:
- Perpendicular bisectors of the sides: Circumcenter
- Bisectors of the angles: Incenter
- Medians: Centroid
- Lines containing the altitudes: Orthocenter
To find the points of concurrency in a triangle, we need to identify different types of lines and their corresponding points.
1. Perpendicular bisectors: These are lines that pass through the midpoint of each side of the triangle and are perpendicular to those sides. The point of concurrency for the perpendicular bisectors is called the circumcenter.
2. Angle bisectors: These are lines that divide each angle of the triangle into two equal angles. The point of concurrency for the angle bisectors is called the incenter.
3. Medians: These are lines that connect each vertex of the triangle to the midpoint of the opposite side. The point of concurrency for the medians is called the centroid.
4. Altitudes: These are lines that are perpendicular to a side of the triangle and pass through the opposite vertex. The point of concurrency for the altitudes is called the orthocenter.
So, the respective names of the points of concurrency are:
1. Perpendicular bisectors: Circumcenter
2. Angle bisectors: Incenter
3. Medians: Centroid
4. Altitudes: Orthocenter
To find the respective names of the points of concurrency in a triangle, we need to understand the different lines and their properties:
1. Perpendicular Bisectors of the Sides: The perpendicular bisectors of the sides are lines that pass through the midpoint of each side of the triangle and are perpendicular to those sides. These lines intersect at a point called the circumcenter. The circumcenter is the center of the circumcircle, the circle that passes through all three vertices of the triangle.
2. Angle Bisectors: The angle bisectors are lines that divide each angle of the triangle into two equal parts. The angle bisectors of a triangle intersect at a point called the incenter. The incenter is the center of the incircle, the circle that is tangent to all three sides of the triangle.
3. Medians: The medians of a triangle are lines that connect each vertex with the midpoint of the opposite side. The medians intersect at a point called the centroid. The centroid divides each median into two segments, with the segment joining the centroid and the vertex being twice as long as the segment joining the centroid and the midpoint of the opposite side.
4. Altitudes: The altitudes of a triangle are lines drawn from each vertex perpendicular to the opposite side. The altitudes intersect at a point called the orthocenter. The orthocenter may lie inside, outside, or on the triangle, depending on the type of triangle.
To summarize, the respective names of the points of concurrency in a triangle are:
- Perpendicular bisectors of the sides intersect at the circumcenter.
- Angle bisectors intersect at the incenter.
- Medians intersect at the centroid.
- Altitudes intersect at the orthocenter.