Class 9 Exam > Class 9 Notes > Theory - To illustrate that the Internal Bisectors of the Angles of a Triangle Concur at a Point
Theory - To illustrate that the Internal Bisectors of the Angles of a Triangle Concur at a Point - Class 9 PDF Download
Objective:
To illustrate that the internal bisectors of the angles of a triangle concur at a point (called the incentre), which always lies inside the triangle.
Related terms
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Incentre- Incentre of a triangle is defined as the point of intersection of the internal bisectors of a triangle. By internal bisectors, we mean the angle bisectors of interior angles of a triangle. Since there are three interior angles in a triangle, there must be three internal bisectors. The intersection point of all three internal bisectors is known as incentre of a circle.
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Incircle- In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle's incentre.
Properties:
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The incentre is one of the triangle's points of concurrency formed by the intersection of the triangle's three angle bisectors.
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These three angle bisectors are always concurrent and always meet in the triangle's interior (unlike the orthocenter which may or may not intersect in the interior). The incentre is the center of the incircle. The incentre is the one point in the triangle whose distances to the sides are equal.
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If the triangle is obtuse, then the incentre is located in the triangle's interior.
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If the triangle is acute, then the incentre is also located in the triangle's interior.
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If the triangle is right, then the incentre is also located in the triangle's interior.
FAQs on Theory - To illustrate that the Internal Bisectors of the Angles of a Triangle Concur at a Point - Class 9
| 1. What is the concept of internal bisectors of angles in a triangle? |  |
Ans. The concept of internal bisectors of angles in a triangle refers to the lines that divide each angle of the triangle into two equal parts. These bisectors start from the vertices of the triangle and meet at a point called the incenter.
| 2. Do the internal bisectors of angles in a triangle always intersect at a single point? | |
Ans. Yes, the internal bisectors of angles in a triangle always intersect at a single point called the incenter. This is a property of triangles and is known as the incenter theorem.
| 3. What is the significance of the incenter in a triangle? | |
Ans. The incenter of a triangle is significant because it is the center of the inscribed circle, also known as the incircle, of the triangle. The incircle touches all three sides of the triangle and has the maximum possible radius among all circles that can be inscribed within the triangle.
| 4. How can we prove that the internal bisectors of angles in a triangle concur at a point? | |
Ans. The internal bisectors of angles in a triangle can be proved to concur at a point by using the concept of angle bisector theorem. By applying this theorem to each angle of the triangle, we can show that the three bisectors meet at a single point, which is the incenter.
| 5. What are some real-life applications of the concept of internal bisectors of angles in a triangle? | |
Ans. The concept of internal bisectors of angles in a triangle has several real-life applications. For example, it is used in navigation and surveying to determine the direction of a point with respect to three known points. It is also used in architecture and engineering to ensure that structures are symmetrical and balanced. Additionally, the incenter of a triangle is used in computer graphics to create realistic shading and lighting effects.
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