Mr. Watkins inquiry his college student to draw a line of symmetry because that a circlewith facility $O$ pictured below:

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Lisa drew the photo below. Is Lisa correct?

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Brad drew the photo below. Is Brad"s picture correct?

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How countless lines of symmetry walk a circle have? Explain.Explain why each line of the contrary divides the one in half.Explain why every line of symmetry for the circle should go v the center.

You are watching: How many lines of symmetry does a circle have


IM Commentary

A circle has actually an infinite number of symmetries. This contrasts with polygonssuch as the triangles and also quadrilaterals taken into consideration in4.G present of symmetry for trianglesand4.G currently of symmetry for quadrilaterals.The one is, in some sense, the most symmetric 2 dimensional figure and it is partly for this reason that that is for this reason familiar. Coins, clock faces, wheels, the image of the full moon in the sky: these space all examples of circles which us encounter top top a continual basis.

This is an instructional task that gives students a opportunity to reason about lines that symmetry and discover the a circle has an one infinite number of lines the symmetry. Even though the concept of an infinite variety of lines is fairly abstract, 4th graders have the right to understand infinity in casual way. Simply as there is always a portion between any two fractions on the number line, over there is always another line through the facility of the one "between" any type of two lines with the facility of the circle. For this reason if you identify a certain number of lines, you have the right to argue the there is constantly at least one more.

In high school, students should return to this job from 2 viewpoints:

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

This task has an speculative GeoGebra worksheet, with the intentthat instructors might use the to an ext interactively show therelevant content material. The file should be taken into consideration a draftversion, and also feedback on that in the comment ar is highlyencouraged, both in regards to suggestions for advancement and for ideason making use of it effectively. The record can be operation via the totally free onlineapplication GeoGebra, or runlocally if GeoGebra has been mounted on a computer.


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Solution

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

If we fold the circle over any kind of line through the facility $O$, climate the components of the one on each side that the heat will match up. One way to create such a line is to choose a allude on the top fifty percent of the circle and also draw the line through that point and the facility $O$. Similar to there space an infinite variety of points ~ above a heat (if friend pick any type of two points, over there is constantly another one in in between them) there are an infinite number of points ~ above the top fifty percent of the circle. Each of these points can be used to attract a line of symmetry. Due to the fact that there are an infinite number of lines through the center, the circle has actually an infinite variety of lines the symmetry.

When the one is folded over a line of symmetry, the components of the one on each side that the line match up. This means that the components of the one on each side that the line must have the same area. Therefore a line of the opposite divides the circle right into two parts with equal area.

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A heat of symmetry for the circle must reduced the circle into two components with same area. Listed below is a snapshot of 2 lines not containing $O$:

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Note the in each case, because that a heat $L$ with the circle the does notcontain the center $O$, the part of the one on the next of $L$ that contains $O$ is bigger than the component of the one on the side of $L$ which does not contain $O$. For this reason these lines cannot be lines of the contrary as any line of symmetry would cut the circle in half.