Question

which figure shows a line tangent to the circle?

Explanation:

Step1: Recall tangent definition

A tangent to a circle is a line that touches the circle at exactly one point, and the radius to the point of tangency is perpendicular to the tangent line (so the triangle formed by the center, the point of tangency, and the external point is a right triangle, satisfying the Pythagorean theorem \(a^2 + b^2 = c^2\) where \(a\) is the radius, \(b\) is the tangent segment, and \(c\) is the segment from the center to the external point).

Step2: Analyze first figure

Radius \(r = 5\), segment from center to external point \(= 10\), tangent segment \(= 5\)? Wait, check Pythagoras: \(5^2 + 5^2 = 25 + 25 = 50\), \(10^2 = 100\). Not equal. Wait, no, maybe the right angle? Wait, the first figure: the line touches the circle at one point? Wait, no, maybe miscalculation. Wait, first figure: radius is 5, the distance from center to the line's point: the triangle has sides 5 (radius), 5 (tangent?), and 10 (hypotenuse). Wait \(5^2 + 5^2 = 50
eq 10^2 = 100\). So not right triangle.

Step3: Analyze second figure

Radius \(r = 3\), segment from center to external point \(= 9\), tangent segment \(= 6\). Check \(3^2 + 6^2 = 9 + 36 = 45\), \(9^2 = 81\). Not equal.

Step4: Analyze third figure

Radius \(r = 3\), segment from center to external point \(= 5\), tangent segment \(= 4\). Check \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\). So this satisfies the Pythagorean theorem, meaning the triangle is right-angled, so the line is tangent (since radius is perpendicular to tangent at point of contact, so the angle between radius and tangent is 90 degrees, forming right triangle). Also, the line touches the circle at exactly one point (the point where the radius meets the line).

Answer:

The third figure (the one with radius 3, segment 5, and tangent segment 4) shows a line tangent to the circle.