Triangle ABC, inscribed in a circle, has AB = 15 and BC = 25. A tangent to the circle is drawn at B, and a line through A parallel to this tangent intersects line BC at D. Find DC.
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To find the length of DC, we need to use the properties of a tangent and a secant of a circle.
Let's start by drawing the problem. We have triangle ABC inscribed in a circle, with AB measuring 15 units and BC measuring 25 units. A tangent is drawn at point B, and a line through A, parallel to this tangent, intersects line BC at point D.
First, let's consider the properties of a tangent to a circle. When a tangent line is drawn to a circle from a point outside the circle, the line is perpendicular to the radius that intersects the point of tangency. In this case, the tangent line is drawn at point B, so it is perpendicular to the radius that intersects B.
Now, let's consider the secant line AD. A secant is a line that intersects a circle in two points. In our case, it intersects the circle at points A and D.
Since the line AD is parallel to the tangent line drawn at point B, we have a transversal. According to the properties of parallel lines, alternate interior angles are congruent. Therefore, angle ABC is congruent to the angle formed at point D.
Now, let's consider the intercepted arcs. The intercepted arc by angle ABC is the arc AC, and the intercepted arc by angle formed at point D is DC.
The measure of an arc subtended by an angle is equal to twice the measure of the angle. So, the measure of arc AC is twice the measure of angle ABC, while the measure of arc DC is twice the measure of the angle formed at point D.
Since the intercepted arcs by angle ABC and the angle formed at point D are congruent, we can conclude that arcs AC and DC are also congruent.
Now, let's find the length of DC. Since arcs AC and DC are congruent and subtend equal angles at the center of the circle, they also have equal lengths.
The total circumference of the circle is 360 degrees. Since arcs AC and DC are congruent, they divide the total circumference equally. Therefore, the length of arc AC is 1/2 of the total circumference, and the length of arc DC is also 1/2 of the total circumference.
We can calculate the length of arc DC by finding 1/2 of the total circumference.
The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
We have the length of AB = 15, which is equivalent to the radius of the circle since AB is the radius.
Substituting the values, C = 2π(15) = 30π.
Now, let's find the length of arc DC. Since arcs AC and DC are congruent, the length of arc DC is also 1/2 of the total circumference.
arc DC = 1/2 * 30π = 15π
Therefore, the length of DC is 15π units.