Скачать или смотреть Class: 9th | Mathematics (FBISE) | Lecture # | Unit #12 | Theorem #12.1.3 |

Class: 9th |
Mathematics (FBISE) |
Lecture # |
Unit #12 |
Theorem #12.1.3 |
Right Bisectors of Triangle |
The Right Bisectors of the sides of a Triangle are concurrent |
Mathematics Science Group |
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In this unit we stated and will proved the following theorems:
• Any point on the right bisector of a line segment is equidistant
from its end points.
• Any point equidistant from the end points of a line segment is on
the right bisector of it.
• The right bisectors of the sides of a triangle are #concurrent.
• Any point on the bisector of an angle is equidistant from its arms.
• Any point inside an angle, equidistant from its arms, is on the bisector of it.
The bisectors of the angles of a triangle are concurrent.
• Right bisection of a line segment means to draw a perpendicular
at the mid point of line segment.
• Bisection of an angle means to draw a ray to divide the given
angle into two equal parts.
Bisectors in a Triangle
#PerpendicularBisector
The perpendicular bisector of a side of a triangle is a line perpendicular to the side and passing through its midpoint.
The three perpendicular bisectors of the sides of a triangle meet in a single point, called the circumcenter . A point where three or more lines intersect is called a point of concurrency. So, the circumcenter is the point of concurrency of perpendicular bisectors of a triangle.
The circumcenter is equidistant from the vertices of the triangle.
The circle drawn with the circumcenter as the center and the radius equal to this distance passes through all the three vertices and is called circumcircle . This is the smallest circle that the triangle can be inscribed in.
The circumcenter lies inside the triangle for acute triangles, on the hypotenuse for right triangles and lies outside the triangle for obtuse triangles. The circumcenter coincides with the midpoint of the hypotenuse if it is an isosceles right triangle.
Proving the Concurrency of the Perpendicular Bisectors of a Triangle
Let’s prove that the three perpendicular bisectors of the sides of a triangle are concurrent which means that they intersect at one point.