Quiz: GeometryFind .

1.	Quiz: GeometryFind .
Answer» Answer Keys. First of all, LET's DRAW TWO new segments. Since two tangents drawn from a point is equal in length, we have $\overline{BC}=\overline{BE}=b$ . So, $\triangle BCE$ is an isosceles right triangle. Since an angle inside an arc is equal to the circumference angle of that arc, we have $\angle BCE=\angle EDF=45^{\circ}$ . So, $\triangle DEF$ is an isosceles right triangle. Solution. Let's MARK some angles on the diagram. Then, we OBSERVE newly made triangles follow this. $\triangle ADE\sim \triangle FEC$ Let $\overline{EF}=x$ . Now the ratio of the corresponding sides is equal to each other. $\overline{AD}:\overline{EF}=\overline{DE}:\overline{CE}$ $\longleftrightarrow a:x=\sqrt{2}x:\sqrt{2}b$ $\longleftrightarrow a:x=x:b$ $\therefore x^2=ab$ $\therefore\boxed{x=\sqrt{ab} }$

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