Class 9 Exam > Class 9 Notes > Theory - To illustrate that Perpendicular Bisectors of the Sides of a Triangle Concur at a Point
Theory - To illustrate that Perpendicular Bisectors of the Sides of a Triangle Concur at a Point - Class 9 PDF Download
Objective:
To illustrate that perpendicular bisectors of the sides of a triangle concur at a point(called the circumcentre) and it falls inside for an acute-angled triangle, on the hypotenuse of right-angled triangle and outside for an obtuse-angled triangle.
Definition
Circumcentre of a triangle is the point of intersection of all the three perpendicular bisectors of the triangle. It is where the "perpendicular bisectors" (lines that are at right angles to the midpoint of each side) meet. The circumcentre of a triangle is equidistant from its vertices and the distance of the circumcentre from each of the three vertices are called circum-radius of the triangle.
Properties:
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All vertices of triangle are equidistant from circumcentre.
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Circumcentre is also the center of circumcircle.
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For acute angled triangle it lies inside the triangle (see fig(a)).
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For obtuse angled triangle it lies outside the triangle (see fig(c)).
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For right angled triangle it lies at the mid-point of hypotenuse (see fig(b)).
FAQs on Theory - To illustrate that Perpendicular Bisectors of the Sides of a Triangle Concur at a Point - Class 9
| 1. What is the definition of a perpendicular bisector? |  |
Ans. A perpendicular bisector is a line or line segment that cuts another line segment into two equal parts at a right angle.
| 2. What are the properties of a perpendicular bisector? | |
Ans. The properties of a perpendicular bisector are:
- It passes through the midpoint of the line segment.
- It is perpendicular to the line segment.
- It divides the line segment into two equal parts.
| 3. What is the significance of the point where perpendicular bisectors of a triangle concur? | |
Ans. The point where perpendicular bisectors of a triangle concur is called the circumcenter. It is the center of the circumcircle, which is a circle passing through all the vertices of the triangle. The circumcenter has an important role in various geometric constructions and calculations involving triangles.
| 4. How can we prove that perpendicular bisectors of the sides of a triangle concur at a point? | |
Ans. One way to prove that perpendicular bisectors of the sides of a triangle concur at a point is by using the concept of the circumcenter. By constructing the perpendicular bisectors for each side of the triangle, we can show that they intersect at a single point, which is the circumcenter. This can be demonstrated using geometric constructions and the properties of perpendicular bisectors.
| 5. Can the perpendicular bisectors of a triangle be parallel? | |
Ans. No, the perpendicular bisectors of a triangle cannot be parallel. If the perpendicular bisectors were parallel, it would mean that they do not intersect and therefore cannot concur at a point. The concur of the perpendicular bisectors is a unique property of triangles and is one of the important characteristics of the circumcenter.
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