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Questions and Answers
Which of the following statements accurately describes the relationship between a median of a triangle and the side it intersects?
Which of the following statements accurately describes the relationship between a median of a triangle and the side it intersects?
- The median is perpendicular to the bisected side, forming a right angle.
- The median divides the triangle into two congruent triangles. (correct)
- The median connects a vertex to the midpoint of the opposite side.
- The median bisects the angle at the vertex it originates from.
The three medians of a triangle intersect at a single point. What is this point called?
The three medians of a triangle intersect at a single point. What is this point called?
- Centroid
- Orthocenter
- Circumcenter (correct)
- Incenter
Which statement is always true about the circumcenter of a triangle?
Which statement is always true about the circumcenter of a triangle?
- It is equidistant from each vertex of the triangle. (correct)
- It is the point of intersection for the angle bisectors.
- It is located inside the triangle.
- It divides each median in a 2:1 ratio.
What is the primary difference between a median and a right bisector in a triangle?
What is the primary difference between a median and a right bisector in a triangle?
If the coordinates of point A are (-3, 4) and point B are (5, -2), what is the midpoint of the line segment AB?
If the coordinates of point A are (-3, 4) and point B are (5, -2), what is the midpoint of the line segment AB?
Line segment AB has a slope of $\frac{2}{3}$. What is the slope of a line that is perpendicular to AB?
Line segment AB has a slope of $\frac{2}{3}$. What is the slope of a line that is perpendicular to AB?
A line segment has endpoints at (2, 5) and (6, 1). What is the equation of the perpendicular bisector of this line segment?
A line segment has endpoints at (2, 5) and (6, 1). What is the equation of the perpendicular bisector of this line segment?
In triangle ABC, the coordinates of A are (-4, 2) and the midpoint of BC is (3, -1). Determine the equation of the median from A to BC.
In triangle ABC, the coordinates of A are (-4, 2) and the midpoint of BC is (3, -1). Determine the equation of the median from A to BC.
Which of the following is NOT a property of a right bisector?
Which of the following is NOT a property of a right bisector?
A triangle has vertices at A(1, 2), B(5, 2), and C(3, 6). Which of the following is the equation of the right bisector of side AB?
A triangle has vertices at A(1, 2), B(5, 2), and C(3, 6). Which of the following is the equation of the right bisector of side AB?
Flashcards
Median
Median
A line from a vertex to the midpoint of the opposite side.
Centroid
Centroid
The point where all three medians of a triangle intersect.
Right Bisector
Right Bisector
A line intersecting a line segment at a 90-degree angle at its midpoint.
Circumcenter
Circumcenter
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Midpoint
Midpoint
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Negative Reciprocal
Negative Reciprocal
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Study Notes
- Medians and Right Bisectors are equations of lines in triangles.
Median
- A median is a line connecting a vertex to the midpoint of the opposite side.
- A vertex is part of the line.
- All three medians meet at the centroid, also known as the center of gravity.
Right Bisector
- A right bisector is a line that intersects a line segment at a 90-degree angle at the midpoint.
- A right bisector may not connect to a vertex.
- All three right bisectors meet at the circumcenter, which is equidistant to each vertex.
Equations of Medians
- Medians go from a vertex to the midpoint of the opposite side.
- To find the points, the median from A passes through, identify Point A and the midpoint of BC.
- To find the midpoint of BC, use the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).
- To find the equation of the median from A, calculate the slope (m) using the coordinates of Point A and the midpoint of BC.
- Use the slope-intercept form (y = mx + b) to find the equation.
Equations of Perpendicular Bisectors
- Right bisectors go through the midpoint of a line segment at 90 degrees.
- Identify the midpoint of AC through which the right bisector of AC passes.
- To find the coordinates of the midpoint of AC, use the midpoint formula.
- The right bisector intersects AC at a 90-degree angle and has a slope that is the negative reciprocal of the slope of AC.
- To find the equation of the right bisector of AC, determine the slope of AC.
- Use the points to find the y-intercept (b) and then the equation of the line.
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Description
Medians connect a triangle's vertex to the midpoint of the opposite side, converging at the centroid. Right bisectors intersect sides at a 90-degree angle at the midpoint, meeting at the circumcenter. To find median equations, identify Point A and the midpoint of BC, then calculate the slope and use the slope-intercept form.
