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High School — Mathematics
Geometry and Spatial Sense

Overview

Core Resources

    Geometry and Spatial Sense

    Strategies

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      Strategies: Advanced Work

      The following activities are suggestions for working with students who are ready for advanced work. We hope that the activities spark ideas and conversations among teachers about useful classroom strategies that can supplement existing curriculum.

      Properties of Circles

      The definition of a circle is the set of points in a plane that are equidistant from the center point. There are many terms and properties associated with circles, such as arcs, chords, and tangents. Challenge your students with questions that combine using these terms with their other geometry skills.


      Problem

      The radius of circle C is 5 inches. If the length of arc AB is inches, find the length of line segment CD and show mathematical support for your solution.

      Method

      This problem seems to gives us too little information, but we can use deductive reasoning to determine the length of CD. First, since we are given the arc length of AB, it makes sense to calculate the circumference of our circle to determine angle ACB. The circumference of this circle is inches. If we take the ratio of arc length to the circumference, , we see that the arc is actually one quarter of our circle. Therefore, the measure of central angle ACB is one-fourth of 360 degrees, or 90 degrees. Since this angle is a right angle, we know that triangle ACB is a right triangle. Line segments AC and CB are radii of the circle, so their lengths are each 5 inches.

      Now there are two ways we can finish solving the problem. First, using the Pythagorean theorem gives us: , so the length of AB is . Since triangle ACB is isosceles, line segment CD is the perpendicular bisector of AB, so we know that line segment DB equals . Using the Pythagorean theorem again with the lengths of line segments DB and BC in place, . We can simplify the equation to see that line segment CD is inches or approximately 3.54 inches.

      Second, since AC and CB are have the same length, angles CAB and CBA are each 45 degrees. Therefore, triangle ACD is a 45-45-90 triangle and the length of each of the legs is . CD is a leg of triangle ACD, so its length is or 3.54 inches.

      Answer

      The length of line segment CD is inches or 3.54 inches.

      Extension

      Ask students to try to make up their own problems that use at least three properties from different areas of geometry.

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