Question - Geometric Problem: Tangent Line and Circle
Solution:
The image contains a geometric problem involving a circle with a tangent line. The problem statement reads:"In the diagram, the center of the circle is $$ O $$ and $$ OT = 23 $$. Calculate $$ DT $$."Let's analyze the diagram. There's a triangle $$ OAT $$ with a right angle at $$ A $$, because the radius of a circle is perpendicular to the tangent at the point of tangency. Given that $$ OT $$ (the radius) is $$ 23 $$ units long, we are dealing with the Pythagorean theorem to find $$ DT $$. Since $$ OA = OT $$ (both are radii of the same circle), $$ OA $$ is also $$ 23 $$ units long. Now we have:\[ OA^2 + AD^2 = OT^2 \]To find $$ DT $$, we need to realize that $$ DT = AD $$. Therefore, we actually need to find $$ AD $$. Given that $$ OA = OT = 23 $$ units, by substituting the values in the Pythagorean theorem, we get:\[ 23^2 + AD^2 = 23^2 \]\[ 529 + AD^2 = 529 \]\[ AD^2 = 529 - 529 \]\[ AD^2 = 0 \]\[ AD = 0 \]Thus, $$ DT = AD = 0 $$. This indicates that point $$ D $$ coincides with point $$ A $$, and the length of segment $$ DT $$ is $$ 0 $$ units.
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