Inscribed Circles in Triangles

This video tutorial teaches how to construct an inscribed circle within a triangle using angle bisectors. It begins with an introduction to angle bisectors and the concept of triangle centers, including the Euler line. The tutorial then defines an inscribed circle and explains its properties, such as the center being the intersection of angle bisectors. The main focus is on the step-by-step construction of the inscribed circle, including drawing the triangle, finding angle bisectors, and determining the circle's center. Finally, it covers how to find the radius of the inscribed circle using a perpendicular line segment.

What is the primary method used to construct an inscribed circle in a triangle?

• Using the triangle's altitudes

• Using the triangle's angle bisectors

• Using the triangle's medians

• Using the triangle's perpendicular bisectors

What is an angle bisector?

• A line that divides a triangle into two equal areas

• A ray that divides an angle into two congruent parts

• A segment that connects the midpoints of two sides of a triangle

• A line that is perpendicular to a side of the triangle

Which of the following is true about the centers of triangles?

• Triangles have centers only if they are equilateral

• Triangles have multiple centers

• All triangles have only one center

• Triangles have no centers

Where is the center of an inscribed circle located?

• At the intersection of the triangle's angle bisectors

• At the intersection of the triangle's medians

• At the centroid of the triangle

• At the midpoint of the triangle's longest side

What is the defining property of an inscribed circle?

• It is centered at the triangle's centroid

• It passes through all three vertices of the triangle

• It is the smallest circle that can fit inside the triangle

• It is tangent to all three sides of the triangle

What is the first step in constructing an inscribed circle?

• Draw the triangle's altitudes

• Draw the triangle

• Draw a circle around the triangle

• Draw the triangle's medians

How do you find the angle bisectors when constructing an inscribed circle?

• By drawing lines parallel to the sides

• By drawing perpendicular bisectors of the sides

• By using the points where a circle intersects the triangle

• By connecting the midpoints of the sides

What is the role of the perpendicular line segment in finding the radius of the inscribed circle?

• It is used to find the centroid of the triangle

• It divides the triangle into two equal areas

• It is used to find the circumcenter of the triangle

• It acts as the radius of the inscribed circle

How is the radius of the inscribed circle determined?

• By calculating the area of the triangle

• By measuring the distance from the center to a vertex

• By drawing a perpendicular from the center to a side

• By measuring the longest side of the triangle

What tool is used to draw the inscribed circle once the center and radius are known?

• A ruler

• A protractor

• A compass

• A set square