Mr. Watkins request his college student to draw a line of symmetry for a circlewith center \$O\$ pictured below:

Lisa drew the picture below. Is Lisa correct?

How many lines the symmetry go a circle have? Explain.Explain why every line of the contrary divides the one in half.Explain why every line of symmetry because that the circle must go with the center.

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IM Commentary

A circle has actually an infinite number of symmetries. This contrasts with polygonssuch as the triangles and quadrilaterals considered in4.G present of symmetry for trianglesand4.G lines of symmetry for quadrilaterals.The circle is, in some sense, the many symmetric 2 dimensional figure and also it is partly for this reason that it is therefore familiar. Coins, clock faces, wheels, the image of the complete moon in the sky: these are all examples of circles which we encounter on a constant basis.

This is one instructional task that gives students a possibility to reason around lines the symmetry and discover that a circle has an one infinite number of lines the symmetry. Even though the ide of an infinite number of lines is fairly abstract, 4th graders have the right to understand infinity in casual way. Just as over there is constantly a portion between any kind of two fountain on the number line, there is constantly another line with the center of the one "between" any type of two lines with the center of the circle. For this reason if you recognize a certain number of lines, you deserve to argue that there is constantly at the very least one more.

In high school, students need to return to this job from two viewpoints:

The algebraic perspective, utilizing the equation that defines a circle, andThe geometric perspective, utilizing the meaning of reflections in terms of perpendicular lines.

This task consists of an experimental GeoGebra worksheet, v the intentthat instructors can use that to much more interactively show therelevant contents material. The record should be taken into consideration a draftversion, and also feedback on that in the comment ar is highlyencouraged, both in terms of suggestions for innovation and for ideason making use of it effectively. The paper can be operation via the totally free onlineapplication GeoGebra, or runlocally if GeoGebra has actually been set up on a computer.

Solution

Lisa is correct. If us fold the circle end the line she has drawn then theparts the the one on each side that the line enhance up.Brad is additionally correct. If us fold the circle over the heat he has drawn then the parts of the one on every side the the line complement up.

If we fold the one over any line through the center \$O\$, then the parts of the one on each side the the heat will enhance up. One method to develop such a heat is to pick a suggest on the top fifty percent of the circle and draw the line through that suggest and the facility \$O\$. Similar to there space an infinite number of points on a line (if girlfriend pick any two points, there is always another one in in between them) there are an infinite number of points ~ above the top fifty percent of the circle. Every of these points can be supplied to draw a line of symmetry. Since there space an infinite number of lines through the center, the circle has actually an infinite number of lines of symmetry.

When the circle is folded over a line of symmetry, the components of the one on each side the the line enhance up. This method that the parts of the circle on every side of the heat must have actually the exact same area. So a line of the contrary divides the circle into two parts with equal area.

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A heat of symmetry because that the one must cut the circle into two parts with same area. Below is a picture of two lines not containing \$O\$:

Note the in every case, because that a line \$L\$ with the circle the does notcontain the facility \$O\$, the component of the circle on the next of \$L\$ that has \$O\$ is larger than the part of the one on the next of \$L\$ which does no contain \$O\$. Therefore these lines can not be present of the opposite as any line of symmetry would cut the circle in half.