If a tangent segment and a secant segment are drawn to a circle from the same point, the external part of the secant segment is longer than the tangent segment. True or False?
True.
When a tangent segment and a secant segment are drawn to a circle from the same point, the external part of the secant segment is longer than the tangent segment. This can be proved using the properties of tangents and secants to a circle.
True.
To understand why the external part of the secant segment is longer than the tangent segment, we can consider the properties of tangents and secants in circles.
A tangent is a line that touches a circle at exactly one point, called the point of tangency. It is always perpendicular to the radius of the circle at the point of tangency. Therefore, a tangent segment is the line segment that starts at the point of tangency and ends at a point outside the circle.
A secant is a line that intersects a circle at two different points. It can be extended to intersect the circle at points on both sides. Therefore, a secant segment is the line segment that starts at the point of intersection outside the circle and ends at another point outside the circle.
Now, when a tangent segment and a secant segment are drawn from the same point to a circle, they will have a common starting point. However, the tangent segment only touches the circle at one point and extends away from the circle, while the secant segment intersects the circle at two points.
Since the secant segment intersects the circle at two points, it must extend beyond the point of tangency where the tangent segment ends. Therefore, the external part of the secant segment is longer than the tangent segment.
Hence, the statement is true.