Question

which figure shows a line tangent to the circle? (three figures with circles, lines, and numbers 6,9,3; 5,10,5; 5,3,4 respectively, and multiple - choice circles below each figure)

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. Also, for a tangent from an external point, the length of the tangent segment, the radius, and the line from the external point to the center form a right triangle (Pythagorean theorem: \(a^2 + b^2 = c^2\) where \(a\) is radius, \(b\) is tangent length, \(c\) is distance from external point to center).

Step2: Analyze first figure

In the first figure, radius is 3, the other segment from center is 9, and the external segment is 6. Check Pythagoras: \(3^2 + 6^2 = 9 + 36 = 45\), \(9^2 = 81\). \(45
eq 81\), so not a right triangle (not tangent, maybe secant).

Step3: Analyze second figure

Radius is 5, the segment from center to the line's point is 5 (so it's a radius, meaning the line passes through two points on the circle? Wait, the line has a segment of 5 from the center, so it's a radius, so the line would be a radius or a chord? Wait, the external segment is 5, and the center-segment is 5, so distance from external point to center is \(5 + 5 = 10\)? Wait, no, the triangle has sides 5, 5, 10? But \(5 + 5 = 10\), which is degenerate (not a triangle, so the line is a radius extended, passing through the circle twice? So it's a secant or diameter, not tangent.

Step4: Analyze third figure

Radius is 3, distance from external point to center is 5, tangent length is 4. Check Pythagoras: \(3^2 + 4^2 = 9 + 16 = 25\), and \(5^2 = 25\). So \(3^2 + 4^2 = 5^2\), which satisfies the right triangle (radius perpendicular to tangent, so the line is tangent as it touches at one point and forms right triangle with radius and external line).

Answer:

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