# Three Proofs

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**Three Proofs**

The Spherical Mirror Formula

Figure 35-19 shows a point object 0 placed on the central axis of a concave spherical mirror. outside its center of curvature C. A ray from 0 that makes an angle a with the axis intersects the axis at / after reflection from the mirror at a. A ray that leaves o along the axis is reflected back along itself at c and also passes through J. Thus. / is the image of 0 it is a real image because light actually passes through it. Let us find the image distance i. A trigonometry theorem that is useful here tells us that an exterior angle of a triangle is equal to the sum of the two opposite interior angles. Applying this to triangles OaC and Oal in Fig. 35-19 yields

Only the equation for 13 is exact, because the center of curvature of arc oc is at C. However, the equations for a and ‘Yare approximately correct if these angles are small enough (that is, for rays close to the central axis). Substituting Eqs. into Eq. 35-16, using Eq. 35-3 to replacer with 2f, and canceling ac lead exactly to Eq.