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Prove theorem 10.1 of circle. ( The tangent at any point of a circle is perpendicular to the radius through the point of contact. )?
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Prove theorem 10.1 of circle. ( The tangent at any point of a circle i...


- Proof of Theorem 10.1: Tangent at any point of a circle is perpendicular to the radius through the point of contact



- Given: A circle with center O, a point P on the circle, and a tangent line at point P.



- Proof:



- Step 1: Draw a radius from the center O to the point of contact P.



- Let OP be the radius and PT be the tangent at point P.



- Step 2: Assume the tangent PT is not perpendicular to the radius OP.



- Let's assume PT is not perpendicular to OP.



- Step 3: Draw a line from point P perpendicular to the radius OP.



- Let this line intersect the radius at point Q.



- Step 4: By definition of perpendicular lines, angle OPT = 90 degrees.



- Since PT is tangent and PQ is perpendicular to OP, angle OPT must be 90 degrees.



- Step 5: This contradicts the assumption that PT is not perpendicular to OP.



- As we assumed PT is not perpendicular to OP, but we have proved that angle OPT is 90 degrees, which is a contradiction.



- Step 6: Therefore, our assumption is false, and PT must be perpendicular to OP.



- Hence, the tangent at any point of a circle is perpendicular to the radius through the point of contact.



- Conclusion: Theorem 10.1 is proved.
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Prove theorem 10.1 of circle. ( The tangent at any point of a circle is perpendicular to the radius through the point of contact. )?
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