Question

17 determine whether \\(\overline{ab}\\) is tangent to the circle with center \\(c\\).
circles\tyes\tno
circle with center c, radius 3 (ca), cb=5, ab=4; then another circle with center c, radius 9 (cb), ca=18, ab=15

Explanation:

Step1: Use the Pythagorean theorem

For a line to be tangent to a circle, it must be perpendicular to the radius at the point of tangency. We can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse of a right - triangle) to check if \(\triangle CAB\) is a right - triangle.
For the first circle:
Let \(CA = 3\), \(AB = 4\), and \(CB=5\).
Substitute into the Pythagorean theorem: \(CA^{2}+AB^{2}=3^{2}+4^{2}=9 + 16=25\), and \(CB^{2}=5^{2}=25\). So \(CA^{2}+AB^{2}=CB^{2}\), which means \(\angle CAB = 90^{\circ}\), and \(\overline{AB}\) is tangent to the circle.

Step2: Use the Pythagorean theorem for the second circle

Let \(CB = 9\), \(AB = 15\), and \(CA=9 + 18=27\).
Substitute into the Pythagorean theorem: \(CB^{2}+AB^{2}=9^{2}+15^{2}=81+225 = 306\), and \(CA^{2}=27^{2}=729\). Since \(CB^{2}+AB^{2}
eq CA^{2}\), \(\angle CBA
eq90^{\circ}\), and \(\overline{AB}\) is not tangent to the circle.

Answer:

For the first circle: Yes. For the second circle: No.