Question

determine if line ab is tangent to the circle.
10)
11)
problem 10
choose
problem 11
choose

Explanation:

Problem 10

Step1: Recall tangent property

A tangent to a circle is perpendicular to the radius at the point of contact. So, if \( AB \) is tangent to the circle at \( A \), then \( \angle BAC = 90^\circ \) (where \( C \) is the center), and triangle \( ABC \) should be a right triangle. We can check using the Pythagorean theorem. The radius \( r = 8 \), so the diameter is \( 16 \)? Wait, no, the length from \( A \) to the other end of the diameter? Wait, the line from \( A \) through the center is length \( 8 \)? Wait, no, the diagram: \( AB = 10 \), the other side from \( B \) to the circle is \( 20 \)? Wait, no, the radius is \( 8 \), so the length from \( A \) to the center is \( 8 \), so the diameter is \( 16 \)? Wait, no, the segment from \( A \) to the other intersection point? Wait, maybe the triangle has sides: \( AB = 10 \), the radius (or the segment from \( A \) to center) is \( 8 \), and the segment from \( B \) to the center? Wait, no, let's re-express. If \( AB \) is tangent at \( A \), then \( OA \perp AB \) ( \( O \) is center). So \( OA = 8 \) (radius), \( AB = 10 \), and \( OB \) is the hypotenuse. Let's calculate \( OB \) if it's a right triangle: \( OB^2 = OA^2 + AB^2 = 8^2 + 10^2 = 64 + 100 = 164 \). But in the diagram, the length from \( B \) to the circle is \( 20 \)? Wait, no, maybe the segment from \( B \) to the other point on the circle is \( 20 \), so \( OB = 20 \)? Wait, that can't be. Wait, maybe I misread. Let's check the Pythagorean theorem: \( 8^2 + 10^2 = 64 + 100 = 164 \), and \( 20^2 = 400 \). Since \( 8^2 + 10^2
eq 20^2 \), so triangle \( OAB \) is not right-angled at \( A \). Therefore, \( AB \) is not tangent.
Wait, maybe the other side: the length from \( B \) to the circle is \( 20 \), so \( OB = 20 \), \( OA = 8 \), \( AB = 10 \). Then \( OA^2 + AB^2 = 64 + 100 = 164
eq 20^2 = 400 \), so \( \angle OAB \) is not \( 90^\circ \), so \( AB \) is not tangent.

Step2: Conclusion

Since \( 8^2 + 10^2
eq 20^2 \) (wait, maybe I got the sides wrong). Wait, maybe the segment from \( B \) to \( A \) is \( 10 \), the segment from \( A \) to the center is \( 8 \), and the segment from \( B \) to the center is \( 20 \)? No, that doesn't make sense. Wait, maybe the triangle has sides \( AB = 10 \), \( OA = 8 \), and \( OB = 20 \)? Then \( 8^2 + 10^2 = 164 \), \( 20^2 = 400 \), not equal. So \( AB \) is not tangent.

Step1: Recall tangent property

If \( AB \) is tangent to the circle at \( A \), then \( OA \perp AB \) ( \( O \) is center). So \( OA = 7.2 \) (radius), \( AB = 9.6 \), and \( OB = 12 \). Let's check Pythagorean theorem: \( OA^2 + AB^2 = 7.2^2 + 9.6^2 \). Calculate \( 7.2^2 = 51.84 \), \( 9.6^2 = 92.16 \). Sum: \( 51.84 + 92.16 = 144 \). And \( OB^2 = 12^2 = 144 \). So \( OA^2 + AB^2 = OB^2 \), so triangle \( OAB \) is right-angled at \( A \). Therefore, \( OA \perp AB \), so \( AB \) is tangent to the circle.

Step2: Verify

\( 7.2^2 + 9.6^2 = (7.2 \times 7.2) + (9.6 \times 9.6) = 51.84 + 92.16 = 144 \), and \( 12^2 = 144 \). So by Pythagorean theorem, \( \angle OAB = 90^\circ \), so \( AB \) is tangent.

Answer:

\( AB \) is not tangent to the circle.

Problem 11