Question

identifying a tangent line
which figure shows a line tangent to the circle?
(three figures with circles, lines, and labeled lengths: first figure has circle with center, triangle with sides 5, 3, and a line; second figure has circle with center, lengths 5, 10, and a line; third figure has circle with center, lengths 6, 9, 3, and a line)

Explanation:

Step1: Recall Tangent Definition

A tangent to a circle touches the circle at exactly one point, and the radius to the point of tangency is perpendicular to the tangent line. Also, the length of a tangent from an external point to a circle is equal, and the tangent is perpendicular to the radius at the point of contact.

Step2: Analyze First Figure

In the first figure, the line at the point of contact: the radius is 5 (distance from center to the point on circle), and the triangle has sides 3,4,5 (3 - 4 - 5 right triangle). But the line at the contact point—check if it's perpendicular to the radius. The radius is 5, the other side is 3, and the tangent - like line: but wait, the triangle with sides 3,4,5 is right - angled (since \(3^{2}+4^{2}=9 + 16=25 = 5^{2}\)). But the line at the contact point: the radius is 5, and the line from the external point to the center is 5? Wait, no, maybe miscalculation. Wait, the first figure: the center to the point on the circle is 5 (radius), the external point to the center is 5? No, the triangle has sides 3,4,5. Wait, maybe not. Let's check the second figure.

Step3: Analyze Second Figure

In the second figure, the radius is 5 (distance from center to the point on the circle). The line from the external point to the center is 10, and the line from the external point to the point on the circle is 5. Wait, but the radius is 5, and the line from external point to the point on the circle is 5, and the radius is 5. So the triangle with sides 5,5,10? But \(5 + 5=10\), which would mean the points are colinear, but that's not possible for a triangle. Also, the line at the point on the circle: the radius is 5, and the line from external point to the point on the circle is 5, so it's not a tangent (since a tangent should be perpendicular to the radius, and here the length from external point to the point on the circle is equal to the radius, which would mean the line is a secant or not a tangent).

Step4: Analyze Third Figure

In the third figure, the radius is 3 (distance from center to the point on the circle). The line from the external point to the center: let's check the lengths. The external point to the point on the circle is 6, and the center to the point on the circle is 3, and the external point to the center is 9? Wait, no, the triangle has sides 3,6,9? No, \(3+6 = 9\), which is colinear. Wait, no, maybe I misread. Wait, the third figure: the radius is 3, the line from the external point to the point on the circle is 6, and the line from the external point to the center is 9? Wait, no, the triangle has sides 3,6,9? But \(3+6=9\), so it's a straight line. That can't be. Wait, going back to the first figure. Wait, the first figure: the center to the point on the circle is 5 (radius), the external point to the center is 5? No, the triangle has sides 3,4,5. So the angle at the point on the circle: since \(3^{2}+4^{2}=5^{2}\), the angle between the radius (length 5) and the line from the external point to the point on the circle (length 4) is 90 degrees? Wait, no, the sides are 3 (from center to the point on the circle? No, wait the center to the point on the circle is 5? Wait, the first figure: the center is a black dot, the point on the circle is another black dot, the distance between them is 5 (radius). Then there is a line from an external point (black dot) to the center (length 5) and to the point on the circle (length 4), and the side between the center and the point on the circle is 3? Wait, that can't be, because the radius should be the distance from center to the circle, so it should be 5. So maybe the first figure is wrong. Wait, the second figure: the radius is 5, the line from the external point to the point on the circle is 5, and the line from external point to center is 10. But \(5^{2}+5^{2}=50 eq10^{2} = 100\), so not right - angled. Wait, the third figure: radius is 3, line from external point to point on circle is 6, line from external point to center is 9. \(3^{2}+6^{2}=9 + 36 = 45 eq9^{2}=81\). Wait, maybe I made a mistake. Wait, the definition of a tangent: a tangent line touches the circle at exactly one point, and the radius to that point is perpendicular to the tangent. Also, the length of the tangent from an external point to a circle is given by \(L=\sqrt{d^{2}-r^{2}}\), where \(d\) is the distance from the external point to the center, and \(r\) is the radius.

Let's re - analyze:

First figure: Let the center be \(O\), the point on the circle be \(A\), and the external point be \(P\). \(OA = 5\) (radius), \(PA = 4\), \(OP=5\). Then by Pythagoras, \(PA^{2}+OA^{2}=16 + 25=41 eq OP^{2}=25\). No. Wait, maybe \(OA = 3\)? Wait, the first figure has a triangle with sides 3,4,5. So if \(OA = 3\) (radius), \(PA = 4\), \(OP = 5\), then \(3^{2}+4^{2}=5^{2}\), so \(\angle OAP = 90^{\circ}\). So \(PA\) is perpendicular to \(OA\) (the radius). So \(PA\) is a tangent line, because the radius is perpendicular to the line at the point of contact (\(A\)).

Second figure: Let \(O\) be the center, \(A\) be the point on the circle, \(P\) be the external point. \(OA = 5\) (radius), \(PA = 5\), \(OP = 10\). \(5^{2}+5^{2}=50 eq10^{2}=100\), so \(\angle OAP\) is not \(90^{\circ}\), so \(PA\) is not a tangent.

Third figure: Let \(O\) be the center, \(A\) be the point on the circle, \(P\) be the external point. \(OA = 3\) (radius), \(PA = 6\), \(OP = 9\). \(3^{2}+6^{2}=9 + 36 = 45 eq9^{2}=81\), so \(\angle OAP\) is not \(90^{\circ}\), so \(PA\) is not a tangent.

So the first figure (the left - most figure) shows a line tangent to the circle because the radius is perpendicular to the line at the point of contact (by the Pythagorean theorem, the triangle is right - angled at the point of contact, so the radius is perpendicular to the line, satisfying the tangent condition).

Answer:

The Left - most Figure