Mr. Watkins request his college student to attract a heat of symmetry because that a circlewith center \$O\$ pictured below:

Lisa attracted the snapshot below. Is Lisa correct?

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

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## IM Commentary

A circle has actually an infinite variety of symmetries. This contrasts v polygonssuch together the triangles and also quadrilaterals considered in4.G currently of symmetry for trianglesand4.G currently of symmetry for quadrilaterals.The one is, in part sense, the most symmetric two dimensional figure and it is partly for this reason that that is so familiar. Coins, clock faces, wheels, the photo of the full moon in the sky: these are all instances of one which us encounter ~ above a constant basis.

This is an instructional job that gives students a opportunity to reason about lines that symmetry and also discover the a circle has actually an one infinite number of lines the symmetry. Even though the ide of an infinite variety of lines is relatively abstract, fourth graders deserve to understand infinity in casual way. Simply as over there is constantly a portion between any type of two fractions on the number line, over there is always another line with the facility of the circle "between" any type of two lines through the facility of the circle. For this reason if you recognize a certain variety of lines, you can argue the there is constantly at least one more.

The algebraic perspective, making use of the equation that specifies a circle, andThe geometric perspective, utilizing the definition of reflect in regards to perpendicular lines.

This task contains an experimental GeoGebra worksheet, v the intentthat instructors can use it to more interactively demonstrate therelevant content material. The document should be taken into consideration a draftversion, and also feedback on that in the comment section is highlyencouraged, both in regards to suggestions for innovation and for ideason utilizing it effectively. The document can be run via the complimentary onlineapplication GeoGebra, or runlocally if GeoGebra has been installed on a computer.

## Solution

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

If us fold the one over any line through the center \$O\$, then the components of the one on every side the the line will match up. One method to create such a line is to choose a suggest on the top half of the circle and also draw the line v that point and the center \$O\$. Just like there room an infinite number of points top top a heat (if girlfriend pick any two points, over there is always another one in between them) there space an infinite number of points top top the top half of the circle. Every of this points have the right to be supplied to draw a heat of symmetry. Because there room an infinite variety of lines with the center, the circle has actually an infinite variety of lines of symmetry.

When the circle is folded end a heat of symmetry, the parts of the one on every side that the line complement up. This way that the parts of the one on every side of the heat must have actually the very same area. So a heat of symmetry divides the circle right into two parts with equal area.

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A heat of symmetry because that the circle must reduced the circle into two parts with equal area. Listed below is a photo of two lines not containing \$O\$:

Note that in every case, for a heat \$L\$ v the circle the does notcontain the facility \$O\$, the part of the circle on the next of \$L\$ that has \$O\$ is larger than the component of the circle on the side of \$L\$ i m sorry does no contain \$O\$. For this reason these lines can not be lines of symmetry as any type of line of the opposite would reduced the circle in half.