ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE

Transcription

1 ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE Keywords: Central angle An angle whose vertex is the center of a circle and whose sides contain the radii of the circle Arc Two points on a circle and the continuous (unbroken) part of the circle between the twp points. The two points are called endpoints Degrees A unit of measure for angles Radians Is an Standard International (SI) unit of plane angular measurement Radius A segment from a point on the sphere of circle to the center. The length of the segment is also called radius Objectives: Materials: 1) Students will be able to determine the length of an arc of a circle given the central angle. 2) Student will be able to convert from radians to degrees. 3) Students will be able to use a macro. 1) Cabri or Geometer Sketch Pad 2) Pencil and paper 3) Lab handout Ohio Standards: 1) Analyze characteristics and properties of two- and three dimensional geometric shapes and develop mathematical arguments about geometric relationships p 310 2) Specify locations and describe special relationships. P 313 3) Use visualization, special reasoning and geometric modeling to solve problem. P 315 (

2 ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE Team Members: File name: This lab was designed with the assumption that students have working knowledge Cabri and/or Geometer Sketch Pad. Question: Given a circle, central angle, and the radius, can the length of a corresponding arc be determined in degrees? Students should already have general knowledge of circles. For instance, students should already know that there are: 1) 360 degrees in a circle and, 2) 2(pi)(r) = 360 degrees (where Pi = 3.14 and r = radians or radius). Setting the radius of a circle r = 1 simplifies the equations that makes it clear that: 1) 1 radian = 180/Pi degrees and, 2) 1 degree = Pi/180 Understanding this relationship allows us to measure the length of arc in degrees, given the circle, central angle, and radius. The following is an illustration of this property. TASKS: 1) Draw a circle (circle tool) 2) Draw two points on the circle (point tool) 3) Label one point A, one B, and center O (label tool) 4) Construct a segment (radius) from (segment tool) each point to the center of the circle. 5) These segments create an angle with vertex at the center of the circle. This is called a Central Angle. 1

3 6) Measure the: a. Central Angle = (angle tool) b. Radius = (distance and length tool) 7) To measure the arc, it is necessary to add a point (pointer tool) between point A and point B. Label this point C. 8) The arc must be marked so the computer will (arc tool) recognize it. Mark the arc using points A, C, and B. 9) Measure the arc = (length and distance tool) If you are using Cabri, notice that the arc measurement is given in centimeters. We will use our knowledge of circles to convert this measurement from centimeters to degrees. The following steps will guide you through this process. 1) Take the arc measurement (in centimeters) Divide it by the measurement (in centimeters) of the radius. ACB (centimeters) / Radius = arc (radians) (calculator tool) Drag you results to your workspace and label ACB (radians) (Note: 1 Radian = 180 degrees/pi) 2) Take the arc measurement (in radians) and multiply it by 360 degrees. Then, divide that product by (2* Pi). ACB (radians) * 360 / (2 * Pi) = arc (degrees) (calculator tool) ACB (radians) * 180 / Pi = arc (degrees) Drag the result from the calculator tool and place it by the arc. Use comment tool to rename ACB in degrees. 2

4 Now you are capable of converting arc measurements in centimeters to an arc measurement in degrees. With this knowledge, you can create a macro. To create the macro: 1) Click on the macro tool. 2) Select Initial Object. 3) Then click on the complete circle then click on points A, C, and B 4) Click on macro tool 5) Select Final Object 6) Click on number value of the arc measurement in degrees. 7) Click on macro tool and select Define Macro 8) Fill in the Name of the construction Call it - Arc measurement in degrees 9) Fill in Name for final object Call it - Arc measurement in degrees 10) Fill in the Help for this macro Write - Given a circle, radius and an arc defined by three points on the circle, this macro will calculate the arc in degrees Now you can use your new macro to investigate the relationship between an arc of a circle and the corresponding central angle. 1) Draw a circle (circle tool) 2) Draw the diameter of the circle (line tool) 3) Draw a perpendicular line to the diameter (perpendicular line tool) passing through the center of the circle. 4) Draw all four points of intersection (point tool) 5) Label the points of intersection (label tool) ABCD and label the center O. 6) Make a point on the circle P between point A and point B (point tool) 7) Place an arc from point A to point P and point B (arc tool) 8) Measure AOB = (angle tool) 9) Use your macro to measure the APB (macro) 3

5 Using the same circle, construct, measure, and label five central angles. Then, use your macro to measure the corresponding arc Angle Measurement Arc Measurement With your expanded knowledge, what is the measure of BOD? = Also, with your macro, what is the measure of the corresponding arc? = In radians, what is the measurement of this arc? Conclusions: Write you conjecture here. What can you say about any central angle of a circle and its corresponding arc? Extensions: Your math teacher decided to have a pizza party in your honor. The teacher announces that any student interested in a piece of pizza must first answer a challenging geometry question. 1) Suppose you were really hungry. Without looking at the 16 pizza which would you prefer? a. A piece of pizza that had a corresponding arc = 5 radians b. A piece of pizza that has a corresponding arc = 287 degrees c. A piece of pizza that has an arc measurement of 45 inches 4