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Prove Theorem 9.10 That is, show that if a graph in the plane has any two of three types of symmetry, namely, symmetry about the x-axis, symmetry about the y-axis, and symmetry about the origin, then it has the third type of symmetry as well.

Short Answer

Expert verified

Proof: cis symmetric to awith respect to the origin.

Step by step solution

01

Given information

It has the third type of symmetry as well.

02

Calculation

Take a point a in the first quadrant.

Draw a line l1from the orlgin to aand let the angle formed by l1and the xaxis.

Now reflect the point a about x axis and call it as point b

Draw a line L2 from the origin to b and call β the angle formed by L2 and the x axis.

And call ρ the angle formed by l2 and the y axis.

Then θ=β and l1=l2

03

Calculation

Now reflectb about y axis and call that paint c

Draw l3from the origin to cand call ϕthe angle formed by l3and y-axis.

Then I2=I3and therefore I3=l1

Since x,yaxes are orthogonal, θand βare complements of ϕand ρ

Therefore, θ+β+ϕ+β=180,l1+l3the diameter of the circle with the origin as the center. Therefore, cis symmetric to awith respect to origin.

Hence the proof.

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