Two segments tangent to a circle from the same external point are __________
Congruent, meaning the same length.
Two segments tangent to a circle from the same external point are equal in length.
To understand why this is the case, we can consider the properties of tangents to a circle.
A tangent to a circle is a line that touches the circle at exactly one point. If we draw two tangents from the same external point, they will be completely outside the circle and will intersect it at two different points. These points of intersection are called the points of tangency.
Since the tangents touch the circle at exactly one point, they are perpendicular to the radius of the circle at that point. The radius is a line segment from the center of the circle to any point on the circle.
Now, let's look at the triangle formed by the center of the circle, the point of tangency, and the external point. This triangle is a right triangle, with the radius acting as the hypotenuse.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, the length of the hypotenuse is the radius of the circle, and the lengths of the other two sides are the distances from the center of the circle to the external point and from the point of tangency to the external point.
Since the two tangents are drawn from the same external point, the distances from the center of the circle to the external point are equal. Thus, the lengths of the two sides of the triangle are equal. As a result, the two tangents, which are the sides of the triangle, are equal in length.