Lesson 5

In this activity, students recall that points on perpendicular bisectors of segments are the same distance from each endpoint of the segment. In the next activity, students will construct the remaining perpendicular bisectors of this triangle and then construct the triangle’s circumscribed circle.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Display this image for all to see. Tell students that the dashed line is the perpendicular bisector of segment \(AB\).

For each of the points \(E\), \(F\), and \(G\), ask students if the point is closer to \(A\) or closer to \(B\), and ask how they know. (Point \(F\) is closer to \(A\). Point \(G\) is closer to \(B\). Point \(E\) is the same distance from \(A\) as it is to \(B\). All points to the left of the perpendicular bisector are closer to \(A\), points to the right are closer to \(B\), and points on the bisector are equidistant from \(A\) and \(B\).)

Students construct the circumcenter of the triangle from the previous activity, or the point where the perpendicular bisectors of the sides of the triangle intersect. They determine that the circumcenter is equidistant from the vertices of the triangle based on the properties of perpendicular bisectors. Finally, they construct the circumscribed circle, observing that this is possible because of the equal distances already noted.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Due to the precision of the tools, students’ circles may not pass through all 3 vertices. Explain that if we had more precise tools, the distances would be the same and the circle would pass through all the vertices.

The goal is to conclude that all triangles have a circumscribed circle.

• “What kind of circle did you construct in relation to the triangle?” (It is a circumscribed circle.)
• “We saw that some quadrilaterals have a circumscribed circle, but some do not. Do all triangles have a circumscribed circle, or only some?” (All triangles have a circumscribed circle, because what we did for triangle \(ABC\) could be done for any triangle. We used properties of perpendicular bisectors, not anything special about this triangle.)
• “What other special lines in triangles have we seen that all meet at a central point?” (In the Coordinate Geometry unit, we saw that the medians and the altitudes in a triangle also have single points of intersection.)

Tell students that the point where all 3 perpendicular bisectors of a triangle intersect is called the triangle’s circumcenter. Ask students to add this theorem to their reference charts as you add it to the class reference chart:

The 3 perpendicular bisectors of the sides of a triangle meet at a single point, called the triangle’s circumcenter. This point is the center of the triangle’s circumscribed circle. (Theorem)

If time permits, ask: “For the triangle, once we drew 2 perpendicular bisectors, we knew the third would pass through the point of intersection of the first 2. Why doesn’t a similar argument work for a quadrilateral? Use a general quadrilateral \(ABCD\) as an example.” (Draw the perpendicular bisectors of segments \(AB\) and \(BC\) and call their point of intersection \(P\). This point is equidistant from points \(A,B,\) and \(C\). In a triangle, there are only 3 vertices, so that covers all of them. However, now we have 4 vertices. We have no way to guarantee that point \(P\) is equidistant from point \(D\).)

In this activity, students compare the circumcenters of acute, obtuse, and right triangles. They find that the circumcenter of a right triangle lies on its hypotenuse, the circumcenter of an acute triangle lies inside the triangle, and the circumcenter of an obtuse triangle lies outside the triangle.

This activity works best when each student has access to devices that can run the GeoGebra applet because students will benefit from seeing the relationship in a dynamic way.

If students struggle to get started, ask them to look back at their work from the previous activity.

The purpose of the discussion is to make informal observations about the locations of the circumcenters. Display this applet for all to see. Don’t immediately move the slider that controls the measure of one of the triangle’s angles.

Alternatively, display these images for all to see.

Ask students why it makes sense that the right triangle has its circumcenter on one of the triangle’s sides (we know that a right angle is inscribed in a half circle, so when you draw the hypotenuse of the triangle, it’s the diameter of the circle).

If using the applet, move the slider to show how the position of the circumcenter changes as the angle moves between being obtuse and acute. Ask students to observe and explain what happens to the circumcenter. (When the angle measure is greater than 90 degrees, the arc in which the angle is inscribed grows to more than 180 degrees. The triangle therefore takes up less than half the circle, so the circumcenter must be outside the circle. Alternatively, when the angle gets smaller, the circle “shrinks” to fit around the triangle. The arc in which the angle is inscribed measures less than 180 degrees, so the triangle fits more snugly in the circle and the circumcenter is inside the triangle.)