Perpendicular bisector
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.
Here, is the circumcenter of .
The circumcenter is equidistant from the vertices of the triangle. (See circumcenter theorem.) That is, . 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.
Example 1:
Natha, Hiren and Joe’s homes represent three non-collinear points on a coordinate plane. If they want to meet at a common place such that each one will have to travel the same distance from their homes, how will you decide the meeting point?
Since the points representing the homes are non-collinear, the three points form a triangle.
Now, if you consider the circumcenter of the triangle, it will be equidistant from the vertices. That is, if the circumcenter of the triangle formed by the three homes is chosen as the meeting point, then each one will have to travel the same distance from their home.
Example 2:
Find the value of .
Here, is the point of concurrency of the three perpendicular bisectors of the sides of .
So, is the circumcenter of the triangle.
The circumcenter is equidistant from the vertices. Then, .
That is, .
Solve for .
Angle bisector
The angle bisector of an angle of a triangle is a straight line that divides the angle into two congruent angles.
The three angle bisectors of the angles of a triangle meet in a single point, called the incenter .
Here, is the incenter of .
The incenter is equidistant from the sides of the triangle. That is, . The circle drawn with the incenter as the center and the radius equal to this distance touches all three sides and is called incircle or the inscribed circle of the triangle. This circle is the largest circle that will fit inside the triangle.
For an equilateral triangle the incenter and the circumcenter will be the same.
Example 3:
Misty has a triangular piece of backyard where she wants to build a swimming pool. How can she find the largest circular pool that can be built there?
The largest possible circular pool would have the same size as the largest circle that can be inscribed in the triangular backyard. The largest circle that can be inscribed in a triangle is incircle. This can be determined by finding the point of concurrency of the angle bisectors of each corner of the backyard and then making a circle with this point as center and the shortest distance from this point to the boundary as radius.
Example 4:
Find the length .
Here, is the point of concurrency of the three angle bisectors of and therefore is the incenter. The incenter is equidistant from the sides of the triangle.That is, .
We have the measures of two sides of the right triangle , so it is possible to find the length of the third side.
Here, is the point of concurrency of the three angle bisectors of and therefore is the incenter. The incenter is equidistant from the sides of the triangle.That is, .
We have the measures of two sides of the right triangle , so it is possible to find the length of the third side.
Use the Pythagorean Theorem to find the length .
Since , the length also equals units.
FAQs
In a triangle, if the interior point is equidistant from the two sides of a triangle, then that point lies on the angle bisector of the angle formed by the two line segments. The perpendicular bisector bisects the given line segment into two equal parts, to which it is perpendicular.
Is bisector always 90 degrees? ›
And, a bisector is a line that divides a line into two equal halves. Thus, a perpendicular bisector of a line segment AB implies that it intersects AB at 90 degrees and cuts it into two equal halves.
What are the angle bisector of a triangle answer? ›
Angle Bisector of a Triangle
In a triangle, the angle bisector of an angle is a straight line that divides the angle into two equal or congruent angles. There can be three angle bisectors in every triangle, one for each vertex. The point where these three angle bisectors meet in a triangle is known as its incenter.
Where do the three angle bisectors of a triangle intersect at responses? ›
The three angle bisectors of a triangle intersect at a single point. The point of concurrency of the angle bisectors is called the incenter. The three altitudes of a triangle are concurrent. The point of concurrency is called the orthocenter.
How to solve bisector? ›
Step 1: Take a look at the given figure and identify the bisected angle given. Step 2: Find the measure of the other bisected angle. The other bisected angle will be equal to the angle given. Step 3: Add the measure of the two bisected angles to get the measure of the whole angle.
What is the bisector formula? ›
According to the angle bisector theorem, PQ/PR = QS/RS or a/b = x/y. An angle bisector is a line or ray that divides an angle in a triangle into two equal measures.
What is the rule of angle bisector in triangle? ›
The Angle Bisector Theorem states that when an angle in a triangle is split into two equal angles, it divides the opposite side into two parts. The ratio of these parts will be the same as the ratio of the sides next to the angle.
What is the right bisector of a triangle? ›
Answer and Explanation: Right Bisector: To bisect means to divide in two equal parts. A right bisector of a line segment, hence is another line segment, line or ray which divides the segment into two equal halves, and intersects the segment at a right angle, that is, at 90 degrees.
How to similar triangles? ›
Two triangles are similar if they meet one of the following criteria. : Two pairs of corresponding angles are equal. : Three pairs of corresponding sides are proportional.
What is the formula for the length of the bisector of a triangle? ›
The length of the angle bisector of a standard triangle such as AD in figure 1.1 is AD2 = AB · AC − BD · DC, or AD2 = bc [1 − (a2/(b + c)2)] according to the standard notation of a triangle as it was initially proved by an extension of the angle bisector up to the circumcircle of the triangle.
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.
Are all medians in a triangle equal? ›
The medians of congruent triangles are equal because the corresponding parts of congruent triangles are congruent. In a scalene triangle, the medians have different lengths. In an equilateral triangle, the length of the medians is the same. The centroid of a triangle divides each median in the ratio 2:1.
Do angle bisectors always intersect inside a triangle? ›
It is a theorem in Euclidean geometry that the three interior angle bisectors of a triangle meet in a single point.
What is a triangle with no equal sides? ›
A scalene triangle is a triangle with all three sides of different lengths. In a scalene triangle, there are no equal side lengths and no equal angle measurements, which means the sides and angles are not congruent.
What is the bisector of the sides of a triangle? ›
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 .
How do you find the internal bisector of a triangle? ›
Internal Angle Bisector Theorem Proof. Theorem 1 : For the triangle ΔABC, we can say that AD is the internal bisector for the ∠BAC which intersects BC at point D. According to the theorem we need to prove that;BDDC=ABAC. Step 1 : Mark a line CE that is parallel to AD mathematically saying CE ∥ DA.
How do you identify an angle bisector? ›
Angle bisector in geometry refers to a line that splits an angle into two equal angles. Bisector means the thing that bisects a shape or an object into two equal parts. If we draw a ray that bisects an angle into two equal parts of the same measure, then it is called an angle bisector.
