How is Constructing a Perpendicular Bisector Similar

How is Constructing a Perpendicular Bisector Similar.

Perpendicular Bisector

Perpendicular bisector
is a line that divides a given line segment exactly into two halves forming xc degrees bending at the intersection point. Perpendicular bisector passes through the midpoint of a line segment. It tin can be synthetic using a ruler and a compass. It makes 90° on both sides of the line segment that is being bisected.

In this commodity, we will discuss the concept of the perpendicular bisector and acquire how to construct information technology. We will also explore the perpendicular bisector of a triangle and its properties. We shall solve a few examples in the end for a better understanding of the concept.

i. What is a Perpendicular Bisector?
2. Perpendicular Bisector Definition
3. How to Construct Perpendicular Bisector?
iv. Perpendicular Bisector of a Triangle
v. Properties of Perpendicular Bisector
6. FAQs on Perpendicular Bisector

What is Perpendicular Bisector?

A perpendicular bisector is a line that bisects a line segment in two equal parts and makes an angle of ninety degrees at the point of intersection. In other words, we tin say that a perpendicular bisector divides a line segment at its midpoint making an angle of 90 degrees. Permit u.s. go through the formal definition of information technology in the side by side department to empathize its significant in a better fashion.

Perpendicular Bisector Definition

A perpendicular bisector is defined every bit a line or a line segment that divides a given line segment into ii parts of equal measurement. ‘Bifurcate’ is the term used to describe dividing equally. Perpendicular bisectors intersect the line segment that they bisect and brand four angles of 90° each on both sides. Perpendicular means a line or a line segment making an bending of 90° with another line or line segment. In the figure shown beneath, the perpendicular bisector bisects the line segment AB into two equal halves.

Perpendicular Bisector

How to Construct Perpendicular Bisector?

Perpendicular bisector on a line segment can be constructed hands using a ruler and a compass. The constructed perpendicular bisector divides the given line segment into two equal parts exactly at its midpoint and makes ii congruent line segments.

Steps for Amalgam Perpendicular Bisector

Follow the steps below to construct a perpendicular bisector of a line segment.

  • Step 1: Describe a line segment XY of any suitable length.
  • Step ii: Take a compass, and with X every bit the eye and with more than one-half of the line segment XY as width, depict arcs above and beneath the line segment.
  • Step 3: Repeat the same step with Y as the center.
  • Step iv: Characterization the points of intersection as ‘P’ and ‘Q’.
  • Step v: Join the points ‘P’ and ‘Q’. The signal at which the perpendicular bisector PQ intersects the line segment XY is its midpoint. Label it equally ‘O’.
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Perpendicular Bisector Construction

Perpendicular Bisector of a Triangle

The perpendicular bisector of a triangle is considered to be a line segment that bisects the sides of a triangle and is perpendicular to the sides. It is non necessary that they should pass through the vertex of a triangle but passes through the midpoint of the sides. The perpendicular bisector of the sides of the triangle is perpendicular at the midpoint of the sides of the triangle. The point at which all the three perpendicular bisectors run into is called the circumcenter of the triangle. At that place tin be iii perpendicular bisectors for a triangle (one for each side). The steps of structure of a perpendicular bisector for a triangle are shown below.

  • Draw a triangle and characterization the vertices equally A, B, and C.
  • With B equally the centre and more than half of BC as radius, draw arcs to a higher place and beneath the line segment, BC. Echo the same process without a change in radius with C as the heart.
  • Label the points of intersection of arcs as X and Y respectively and join them. This is the perpendicular bisector for i side of the triangle BC.
  • Repeat the aforementioned process for sides AB and Ac. All the three perpendicular bisectors brand an angle of 90“ at the midpoint of each side.

The perpendicular bisector of a triangle after construction is shown below.XY, HG, and PQ are the perpendicular bisectors of sides BC, AC, and AB respectively.

Perpendicular Bisector of a Triangle

Perpendicular Bisector Properties

Perpendicular bisectors can bisect a line segment or a line or the sides of a triangle. The important properties of a perpendicular bisector are listed below.

Perpendicular bisector,

  • Divides a line segment or a line into two congruent segments.
  • Divides the sides of a triangle into congruent parts.
  • They make an angle of 90° with the line that is beingness bisected.
  • They intersect the line segment exactly at its midpoint.
  • The indicate of intersection of the perpendicular bisectors in a triangle is called its circumcenter.
  • In an astute triangle, they meet within a triangle, in an obtuse triangle they meet outside the triangle, and in right triangles, they meet at the hypotenuse.
  • Whatsoever point on the perpendicular bisector is equidistant from both the ends of the segment that they bifurcate.
  • Tin be only one in number for a given line segment.

Important Notes on Perpendicular Bisector

  • A perpendicular bisector is a line that divides a given line segment exactly into ii halves forming 90 degrees bending at the intersection point.
  • The perpendicular bisector of a triangle is considered to be a line segment that bisects the sides of a triangle and is perpendicular to the sides.
  • Perpendicular bisector on a line segment can be constructed easily using a ruler and a compass.



Related Articles

Given below is the list of topics that are closely connected to the perpendicular bisector. These topics will likewise give you a glimpse of how such concepts are covered in Cuemath.

  • Perpendicular Bisector Theorem
  • Perpendicular Bisector of a Chord
  • Geometry

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FAQs on Perpendicular Bisector

What is the Perpendicular Bisector in Geometry?

Perpendicular Bisector is a line segment that bisects a straight line segment into ii congruent or equal segments. They divide the line segment exactly at its midpoint. Perpendicular bisector makes ninety° with the line segment it bisects.

How Exercise You Construct Perpendicular Bisector With a Straight Edge and a Compass?

Perpendicular bisector is constructed using a straight ruler and a compass using the following steps:

  • Draw a line segment AB of any length with a straight edge or a ruler.
  • Using a compass, with A as the center and more than one-half of AB every bit radius draw arcs above and beneath the line segment AB.
  • Repeat the higher up step with B as the center.
  • Mark the points of intersection as P and Q.
  • Join the points P and Q with a direct edge.
  • The line connecting the points P and Q is the perpendicular bisector to the given line segment which makes 90° with information technology.

Tin can a Perpendicular Bisector e’er be a Median of a Triangle?

Perpendicular bisector can be a median of a triangle only in the case of an equilateral triangle. Median is a line segment joining the vertex of ane side of the triangle to the midpoint of its opposite side. If the median line segment intersects the opposite side at exactly 90° then we can say that the median is a perpendicular bisector. Therefore, the median of a triangle can be a perpendicular bisector but if it makes 90 degrees with the side opposite to it.

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What are the Properties of Perpendicular Bisector?

Few backdrop of the perpendicular bisector are listed below:

  • Perpendicular bisector divides a line segment into congruent segments.
  • They intersect a line segment or the side of a triangle exactly at its midpoint.
  • Divides the sides of the triangle into two equal parts.
  • Tin be only i in number for a given line segment.
  • Any bespeak on the perpendicular bisector is equidistant from both the ends of the segment that they bisect.
  • They make 90 degrees at the midpoint where it touches the line segment it bisects.

What is the Perpendicular Bisector Theorem?

Perpendicular bisector theorem states that any signal on the perpendicular bisector is always equidistant to both the ends of the line segment to which it is perpendicular.

What is the Difference Between Perpendicular Bisector and Bending Bisector?

Perpendicular bisector divides a line segment into ii equal halves, whereas, bending bisector divides a given angle into 2 congruent angles. For example, a perpendicular bisector to a line segment of measure 10 units makes two line segments of 5 units each, whereas, an angle bisector for a given angle of 60 degrees bisects the angle and makes ii angles of thirty degrees each.

What is the Perpendicular Bisector of a Triangle?

Lines that carve up the sides of the triangle into 2 congruent segments are called perpendicular bisectors of a triangle. There can be three perpendicular bisectors for a triangle. They all meet at a signal called circumcenter. It is non necessary that they pass through the vertex of a triangle to its contrary side’south midpoint. Sometimes the perpendicular bisectors originate from a signal that is abroad from the vertex and intersects the opposite side exactly at its midpoint. In an equilateral triangle, the medians of the triangle are perpendicular bisectors equally they make 90 degrees with their opposite sides.

What are the Properties of Perpendicular Bisector of a Chord?

Perpendicular bisector to a chord:

  • Bisects the chord of a circumvolve.
  • Makes ninety degrees with the chord.
  • Passes through the center of the circle.

How Many Perpendicular Bisector Can Be Synthetic For a Line?

In that location tin be only one perpendicular bisector constructed for a line. This is because there can be only one midpoint for a line. This fact is true because the perpendicular bisectors laissez passer through the midpoint of a line or a line segment.

How is Constructing a Perpendicular Bisector Similar

Source: https://www.cuemath.com/geometry/perpendicular-bisectors/