Question

Describe the steps necessary to construct an inscribed circle in a triangle and contrast this with the steps you take to construct a circumscribed circle ?

Answer 1

To construct an inscribed circle, or incircle, in a triangle, bisect the angles and draw a circle using the incenter as the center and the distance to a side as the radius. To construct a circumscribed circle, or circumcircle, around a triangle, bisect the sides perpendicularly and draw a circle using the circumcenter as the center and the distance to a vertex as the radius.

Step-by-step explanation:

To construct an inscribed circle, or incircle, in a triangle, follow these steps:

1. Draw the triangle.
2. Construct the angle bisectors of the triangle's three angles. These will meet at a single point, called the incenter, which is the center of the incircle.
3. Use a compass to draw a circle with the incenter as the center and the distance to any side of the triangle as the radius. This circle will be tangent to all three sides of the triangle.

To construct a circumscribed circle, or circumcircle, around a triangle, follow a different set of steps:

1. Draw the triangle.
2. Perpendicular bisect each of the triangle's sides. This can be done by drawing a line at a right angle to the midpoint of each side.
3. The point where these perpendicular bisectors intersect is the triangle's circumcenter, which will be the center of the circumcircle.
4. Place the compass on the circumcenter and adjust it to reach any of the triangle's vertices. Draw the circle, which should pass through all three vertices of the triangle.

While both the incircle and circumcircle are related to triangles, they are constructed differently - the incircle is tangent to the sides of the triangle and its radius is determined by the distance from the incenter to a side, while the circumcircle passes through the vertices of the triangle and its radius is determined by the distance from the circumcenter to a vertex.