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What is the perpendicular bisector theorem?
If a point is on the perpendicular bisector of a segment, then it is equidistant, or the same distance, from the endpoints of the segment.
If R and S are two points in the plane the perpendicular bisector of is the set of all points equidistant from R and S.?
The perpendicular bisector of the line segment connecting points R and S is a line that is perpendicular to the segment at its midpoint. Any point on this line is equidistant from R and S, meaning the distance from any point on the bisector to R is the same as the distance to S. This property makes the perpendicular bisector a crucial concept in geometry, particularly in triangle construction and circle definition.
If R and S are two points in the plane the perpendicular bisector of line RS is the set of all points equidistant from R and S?
The perpendicular bisector of a line segment RS is a line that is perpendicular to RS at its midpoint. This line consists of all points that are equidistant from both points R and S. Thus, any point on this bisector will have the same distance to R as it does to S. It serves as a geometric locus of points maintaining this equal distance property.
What are all the characteristics of a perpendicular bisector?
A perpendicular bisector is a line that divides a segment into two equal parts at a 90-degree angle. It has two key characteristics: it is equidistant from the endpoints of the segment it bisects, meaning any point on the bisector is the same distance from both endpoints, and it intersects the segment at its midpoint. Additionally, the slope of the perpendicular bisector is the negative reciprocal of the slope of the original segment.
How are perpendicular bisectors and angle bisectors the same?
In general, they are not. In an isosceles triangle, the perpendicular bisector of the base is the same as the bisector of the angle opposite the base. But the other two perp bisectors are not the same as the angle bisectors. Only in an equilateral triangle is each perp bisector the same as the angle bisector of the angle opposite.
Is median and the altitude of a triangle the same?
The median is a line from a vertex to the midpoint of the opposite line and an altitude is a line from a vertex to the opposite line which is perpendicular to the line. These are NOT the same thing in most triangles. The only time they could be the same is in an equilateral triangle.
In an isosceles triangle what are the altitude and media?
In an isosceles triangle, the altitude from the vertex angle to the base bisects the base and is also the median, as it divides the triangle into two congruent right triangles. This altitude is perpendicular to the base, creating two equal segments. Consequently, in an isosceles triangle, the altitude, median, and angle bisector from the vertex angle to the base are all the same line segment.
How do you draw parallel perpendicular bisector?
Two lines cannot be parallel and perpendicular at the same time.
What is equidistant from the two end points of a line segment?
the middle point * * * * * In 2 dimensions: also any point on line forming the perpendicular bisector of the line segment. In 3 dimensions: the plane formed by the perpendicular bisector being rotated along the axis of the line segment. In higher dimensions: Hyperplanes being rotated along the same axis.
How do you find center of a circle given 3 points on the circle?
You have points A, B, and C. Using a compass and straight edge, find a perpendicular bisector of AB (that is, a line that is perpendicular to AB and intersects AB at the midpoint of AB. Next, find a perpendicular bisector of BC. The two lines you found will meet at the center of the circle.
Does a triangle have any lines of symmetry?
It will if isosceles or equilateral. The first will have a single line: the bases' perpendicular bisector produced through the apex. The second will have 3 in the same way.
If orthocentre and circumcentre of a triangle coincide then is it an equilateral triangle?
Yes. The orthocenter is the intersection of the altitudes; the circumcenter is the intersection of the perpendicular bisectors of the three sides of the triangle. The perpendicular bisector of and altitude to a given side are parallel, so they can coincide at the common center only if they are the same; that means that the opposite vertex is on the perpendicular bisector, so the other two sides are equal. Thus each pair of sides are equal, so the triangle is equilateral.
