Where Can the Perpendicular Bisector of an Acute Triangle Intersect
Perpendicular Bisector
Perpendicular bisector is a line that divides a given line segment exactly into two halves forming 90 degrees at the intersection point. Perpendicular bisector passes through the midpoint of a line segment. It can be constructed using a ruler and a compass. It makes 90° on both sides of the line segment that is being bisected.
| 1. | Perpendicular Bisector Definition |
| 2. | How to Construct Perpendicular Bisector? |
| 3. | Perpendicular Bisector of a Triangle |
| 4. | Properties of Perpendicular Bisector |
| 5. | FAQs on Perpendicular Bisector |
Perpendicular Bisector Definition
A perpendicular bisector is defined as a line or a line segment that divides a given line segment into two parts of equal measurement. 'Bisect' is the term used to describe dividing equally. Perpendicular bisectors intersect the line segment that they bisect and make four angles of 90° each on both sides. Perpendicular means a line or a line segment making an angle of 90° with another line or line segment. In the figure shown below, the perpendicular bisector bisects the line segment AB into two equal halves.
How to Construct Perpendicular Bisector?
Perpendicular bisector on a line segment can be constructed easily 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 two congruent line segments.
Steps for Constructing Perpendicular Bisector
Follow the steps below to construct a perpendicular bisector of a line segment.
Step 1: Draw a line segment XY of any suitable length.
Step 2: Take a compass, and with X as the center and with more than half of the line segment XY as width, draw arcs above and below the line segment.
Step 3: Repeat the same step with Y as the center.
Step 4: Label the points of intersection as 'P' and 'Q'.
Step 5: Join the points 'P' and 'Q'. The point at which the perpendicular bisector intersects the line segment XY is its midpoint. Label it as 'O'.
Perpendicular Bisector of a Triangle
Perpendicular bisector of a triangle is considered to be a line segment that bisects the sides of a triangle and are perpendicular to the sides. It is not 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 meet is called the circumcenter of the triangle. There can be three perpendicular bisectors for a triangle (one for each side). The steps of construction of a perpendicular bisector for a triangle are shown below.
- Draw a triangle and label the vertices as A, B, and C.
- With B as the center and more than half of BC as radius, draw arcs above and below the line segment, BC. Repeat the same process without a change in radius with C as the center.
- Label the points of intersection of arcs as X and Y respectively and join them. This is the perpendicular bisector for one side of the triangle BC.
- Repeat the same process for sides AB and AC. All the three perpendicular bisectors make an angle of 90" at the midpoint of each side.
The perpendicular bisector of an equilateral triangle after construction is shown below.XY, HG, and PQ are the perpendicular bisectors of sides BC, AC, and AB respectively.
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 being bisected.
- They intersect the line segment exactly at its midpoint.
- The point of intersection of the perpendicular bisectors in a triangle is called its circumcenter.
- In an acute triangle, they meet inside a triangle, in an obtuse triangle they meet outside the triangle, and in right triangles, they meet at the hypotenuse.
- Any point on the perpendicular bisector is equidistant from both the ends of the segment that they bisect.
- Can be only one in number for a given line segment.
☛ Topics Related to Perpendicular Bisector
Given below is the list of topics that are closely connected to the perpendicular bisector. These topics will also give you a glimpse of how such concepts are covered in Cuemath.
- Perpendicular Bisector Theorem
- Perpendicular Bisector of a Chord
- Geometry
Perpendicular Bisector Examples
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Example 1: Find at which point a perpendicular bisector bisects a line segment of length 10 units.
Solution:
A perpendicular bisector is a line that bisects a given line segment into two congruent line segments exactly at its midpoint. It is given that the line segment is of the length 10 units. So, the perpendicular bisector bisects the line segment exactly at 5 units and the line segment of 10 units is divided into two line segments of 5 units each. -
Example 2: Can you find if the points of intersection of all the perpendicular bisectors for an obtuse triangle PQR with measurements as follows: PQ = 5 units, QR = 8 units, and PR = 9 units lies outside or inside the triangle?
Solution: The perpendicular bisector of any triangle bisects the sides at its midpoint. In a triangle, there are three perpendicular bisectors that can be drawn from each side. To find the perpendicular bisector of a triangle with the given sides, follow the steps given below.
- Draw a line segment PQ of length 5 units.
- With P as the center and more than half of PQ as radius, draw arcs above and below the line segment PQ. Repeat the same process with Q as the center. Join the points of intersection of these arcs.
- Repeat the same process and draw perpendicular bisectors for the sides QR and PR.
The construction of the perpendicular bisectors of the obtuse triangle is shown below.
We can clearly see that all the perpendicular bisectors bisect at a point outside the triangle.
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Example 3: Draw a perpendicular bisector to the diameter of a circle whose radius is 4 units.
Solution:
Given, the radius of the circle = 4 units. Diameter = 2 × radius. So, diameter = 2 × 4 = 8 units. The steps to construct a perpendicular bisector on a diameter of a circle are as follows.- Draw a line segment XY of length 8 units.
- Using a compass, with X as the center and the radius as more than 4 units, draw arcs above and below the line segment.
- Repeat the same process with Y as the center. Label the points of intersection of the arcs as P and Q.
- Join the points P and Q. This line is the perpendicular bisector for the diameter of the circle. Label the point of intersection of the perpendicular bisector with the diameter as O.
The construction discussed in the above steps is shown in the figure below.
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Perpendicular Bisector Practice Questions
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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 two congruent or equal segments. They divide the line segment exactly at its midpoint. Perpendicular bisector makes 90° with the line segment it bisects.
How Do You Construct Perpendicular Bisector With a Straight Edge and a Compass?
Perpendicular bisector is constructed using a straight edge 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 half of AB as radius draw arcs above and below the line segment AB.
- Repeat the above step with B as the center.
- Mark the points of intersection as P and Q.
- Join the points P and Q with a straight edge.
- The line connecting the points P and Q is the perpendicular bisector to the given line segment which makes 90° with it.
Can a Perpendicular Bisector always 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 one 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 only if it makes 90 degrees with the side opposite to it.
What are the Properties of Perpendicular Bisector?
Few properties 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.
- Can be only one in number for a given line segment.
- Any point 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 point 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 Angle Bisector?
Perpendicular bisector divides a line segment into two equal halves, whereas, angle bisector divides a given angle into two 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 two angles of 30 degrees each.
What is the Perpendicular Bisector of a Triangle?
Lines that divide the sides of the triangle into two congruent segments are called perpendicular bisectors of a triangle. There can be three perpendicular bisectors for a triangle. They all meet at a point called circumcenter. It is not necessary that they pass through the vertex of a triangle to its opposite side's midpoint. Sometimes the perpendicular bisectors originate from a point that is away from the vertex and intersects the opposite side exactly at its midpoint. In an equilateral triangle, the medians of the triangle are perpendicular bisectors as 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 circle.
- Makes 90 degrees with the chord.
- Passes through the center of the circle.
How Many Perpendicular Bisector Can Be Constructed For a Line?
There can 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 pass through the midpoint of a line or a line segment.
Where Can the Perpendicular Bisector of an Acute Triangle Intersect
Source: https://www.cuemath.com/geometry/perpendicular-bisectors/