which figure shows a line tangent to the circle?

Answer

Explanation:

Step1: Recall tangent - radius property

If a line is tangent to a circle, then the radius drawn to the point of tangency is perpendicular to the tangent line. By the Pythagorean theorem, in a right - triangle (formed by the radius, the tangent, and the line from the center of the circle to an external point), if (a) is the radius, (b) is the length of the tangent segment from the external point to the point of tangency, and (c) is the length of the line from the center of the circle to the external point, then (a^{2}+b^{2}=c^{2}).

Step2: Check the first figure

For the first figure, if the radius (a = 3), one side (b = 6) and the other side (c = 9). Then (a^{2}+b^{2}=3^{2}+6^{2}=9 + 36=45) and (c^{2}=9^{2}=81). Since (a^{2}+b^{2}\neq c^{2}), the line is not tangent.

Step3: Check the second figure

For the second figure, if the radius (a = 5), one side (b = 5) and the other side (c = 10). Then (a^{2}+b^{2}=5^{2}+5^{2}=25 + 25 = 50) and (c^{2}=10^{2}=100). Since (a^{2}+b^{2}\neq c^{2}), the line is not tangent.

Step4: Check the third figure

For the third figure, if the radius (a = 3), one side (b = 4) and the other side (c = 5). Then (a^{2}+b^{2}=3^{2}+4^{2}=9 + 16=25) and (c^{2}=5^{2}=25). Since (a^{2}+b^{2}=c^{2}), the line is tangent.