Question

which figure shows a line tangent to the circle?

Explanation:

Step1: Recall tangent - radius property

A line is tangent to a circle if and only if it is perpendicular to the radius at the point of tangency. By the Pythagorean theorem, in a right - triangle formed by the radius, the tangent, and the line from the center of the circle to the external point of the tangent, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the distance from the center of the circle to the external point and \(a\) is the radius and \(b\) is the length of the tangent segment.

Step2: Check first figure

For the first figure, if the radius \(r = 3\), the other two sides are \(6\) and \(9\). Using the Pythagorean theorem \(3^{2}+6^{2}=9 + 36=45
eq9^{2}=81\), so the line is not tangent.

Step3: Check second figure

For the second figure, if the radius \(r = 5\), the other two sides are \(5\) and \(10\). Using the Pythagorean theorem \(5^{2}+5^{2}=25 + 25 = 50
eq10^{2}=100\), so the line is not tangent.

Step4: Check third figure

For the third figure, if the radius \(r = 3\), and the other two sides are \(4\) and \(5\). Using the Pythagorean theorem \(3^{2}+4^{2}=9 + 16=25=5^{2}\). So the line is perpendicular to the radius at the point of intersection with the circle, and the line is tangent.

Answer:

The third figure shows a line tangent to the circle.