Which of the following lines intersect to
make the centre of the largest circle that can be
inscribed in the triangle?
Answers
Answer:
Given a triangle, an inscribed circle is the largest circle contained within the triangle. The inscribed circle will touch each of the three sides of the triangle in exactly one point. The center of the circle inscribed in a triangle is the incenter of the triangle, the point where the angle bisectors of the triangle meet.
To construct the inscribed circle:
Construct the incenter.
Construct a line perpendicular to one side of the triangle that passes through the incenter. The segment connecting the incenter with the point of intersection of the triangle and the perpendicular line is the radius of the circle.
Construct a circle centered at the incenter with the radius found in step 2.
The steps for constructing the inscribed circle for a given triangle will be explored in the problems below.
Given a triangle, the circumscribed circle is the circle that passes through all three vertices of the triangle. The center of the circumscribed circle is the circumcenter of the triangle, the point where the perpendicular bisectors of the sides meet.
To construct the circumscribed circle:
Construct the circumcenter.
Construct a circle centered at the circumcenter that passes through one of the vertices. This same circle should pass through all three vertices.
The steps for constructing the circumscribed circle for a given triangle will be explored in the Examples section.Constructing Angle Bisectors
Draw a triangle. Construct the angle bisectors of two of its angles. Why is the point of intersection of the two angle bisectors the incenter of the circle?
Use your compass and straightedge to construct the angle bisector of one of the angles.
Repeat with a second angle.
The point of intersection of the angle bisectors is the incenter. It is not necessary to construct all three angle bisectors because they all meet in the same point. The third angle bisector does not provide any new information.
Constructing Perpendicular Lines
Construct a line perpendicular to one side of the triangle that passes through the incenter of the triangle.
Use your compass and straightedge to construct a line perpendicular to one side of the triangle that passes through the incenter.
Constructing Inscribed Circles
Construct a circle centered at the incenter that passes through the point of intersection of the side of the triangle and the perpendicular line from the problem above.
Note that this circle touches each side of the triangle exactly once.
Answer:
The incenter of the triangle, or the place where the triangle's angle bisectors meet, is the centre of the circle that is encircled by the triangle.
Explanation:
Step : 1 It is the biggest circle that is completely enclosed by a triangle. The intersection of the triangle's three angle bisectors is where the triangle's centre, or incenter, is located. The angle's bisector is the line passing through that point and the vertex. Only two angle bisectors are required to create the inscribed circle of a triangle; the point of intersection between them will serve as the circle's centre.
Step : 2 A diameter is a line segment that traverses a circle by going through its centre. The radius is twice as long as the diameter. The greatest circle that may be created around a spherical is known as a great circle. Great circles exist on all spheres. A sphere would be split perfectly in half if you cut it at one of its great circles. The centre point and circumference of a large circle are identical to those of a sphere.
Step : 3 Inscribe a circle in the given triangle.
Step 1: We draw angle bisectors for 2 angles and mark their intersection.
Step 2: Next, we drop a perpendicular line from the incenter of the circle to one edge of the triangle.
Step 3: Finally, we construct a circle where the perpendicular line from Step 2 is the radius.
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