Question

two tangents each intersect a circle at opposite endpoints of the same diameter. is it possible for the two tangents to intersect each other outside the circle? explain why or why not, using the information you learned in this lesson.

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. Let the circle have center \(O\) and diameter \(AB\). Let the two tangents be \(l_1\) at point \(A\) and \(l_2\) at point \(B\). The radius \(OA\perp l_1\) and \(OB\perp l_2\), and \(OA\) and \(OB\) are collinear (since \(AB\) is a diameter).

Step2: Analyze the orientation of the tangents

Since \(OA\perp l_1\) and \(OB\perp l_2\) and \(OA\) and \(OB\) are on the same straight - line (the diameter), \(l_1\) and \(l_2\) are parallel to each other.

Step3: Determine intersection possibility

Parallel lines do not intersect. So, two tangents that intersect a circle at opposite endpoints of the same diameter cannot intersect each other outside the circle.

Answer:

No, it is not possible. The two tangents are parallel because each is perpendicular to a radius and the two radii are collinear (being part of the same diameter), and parallel lines do not intersect.