why is the tangent to a circle at right angles to the radius?

two methods for establishing this both involve an idea of limit

one way is to establish (RHS) that if you join the middle of a chord to the centre of the circle the line is perpendicular to the chord and then move this chord outwards (parallel to itself) until it just about leaves the circle...

another involves using a chord, extended beyond the circumference at both sides

you can easily show that the two angles that the chord makes with the radiuses are equal (RHS again)

so the supplements (other angle on the straight line) to these angles are equal

again, moving the chord steadily out of the circle shows that these two angles become 90 degrees when the chord becomes a tangent

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