This question tests understanding of tangent lines from an external point to a circle and their geometric properties. The relationship between the tangent length (t), radius (r), and distance from center to external point (d) forms a right triangle where t² + r² = d² by the Pythagorean theorem. Given that the circle has radius r = 7 and the external point P is at distance d = 25 from center O, we form a right triangle with the radius as one leg, the tangent segment as the other leg, and OP as the hypotenuse, giving us t² + 7² = 25², so t² + 49 = 625, thus t² = 576 and t = √576 = 24. Choice A is correct because it applies the Pythagorean theorem correctly with specific values. Choice C incorrectly adds the radius and tangent length instead of using the Pythagorean theorem: d ≠ r + t, rather d² = r² + t². Key to tangent problems: remember that tangent ⊥ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. To find tangent length from external point, identify radius r and distance d to external point, then use t = √(d² - r²); if you recognize a Pythagorean triple, you can determine the answer immediately.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. The relationship between the tangent length (t), radius (r), and distance from center to external point (d) forms a right triangle where t² + r² = d² by the Pythagorean theorem. Given that the circle has radius r = 8 and the external point P is at distance d = 17 from center O, we form a right triangle with the radius as one leg, the tangent segment as the other leg, and OP as the hypotenuse, giving us t² + 8² = 17², so t² + 64 = 289, thus t² = 225 and t = √225 = 15. This is an 8-15-17 Pythagorean triple, so we immediately recognize the tangent length is 15. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice C incorrectly treats the tangent length as the hypotenuse, but actually the distance OP is the hypotenuse. Key to tangent problems: remember that tangent ⊥ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. To find tangent length from external point, identify radius r and distance d to external point, then use t = √(d² - r²); if you recognize a Pythagorean triple, you can determine the answer immediately.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. The relationship between the tangent length (t), radius (r), and distance from center to external point (d) forms a right triangle where t² + r² = d² by the Pythagorean theorem. Given that the circle has radius r = 4 and the external point P is at distance d = 5 from center O, we form a right triangle with the radius as one leg, the tangent segment as the other leg, and OP as the hypotenuse, giving us t² + 4² = 5², so t² + 16 = 25, thus t² = 9 and t = √9 = 3. This is a 3-4-5 Pythagorean triple, so we immediately recognize the tangent length is 3. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice C incorrectly adds the radius and distance instead of using the Pythagorean theorem: d ≠ r + t, rather d² = r² + t². Key to tangent problems: remember that tangent ⊥ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. To find tangent length from external point, identify radius r and distance d to external point, then use t = √(d² - r²); if you recognize a Pythagorean triple, you can determine the answer immediately.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. The relationship between the tangent length ($t$), radius ($r$), and distance from center to external point ($d$) forms a right triangle where $t^2 + r^2 = d^2$ by the Pythagorean theorem. Given that the circle has radius $r = 6$ and the external point $P$ is at distance $d = 10$ from center $O$, we form a right triangle with the radius as one leg, the tangent segment as the other leg, and $OP$ as the hypotenuse, giving us $t^2 + 6^2 = 10^2$, so $t^2 + 36 = 100$, thus $t^2 = 64$ and $t = \sqrt{64} = 8$. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice A incorrectly adds the radius and tangent length instead of using the Pythagorean theorem: $d \neq r + t$, rather $d^2 = r^2 + t^2$. Key to tangent problems: remember that tangent $\perp$ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. To find tangent length from external point, identify radius $r$ and distance $d$ to external point, then use $t = \sqrt{d^2 - r^2}$; if you recognize a Pythagorean triple, you can determine the answer immediately.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. The relationship between the tangent length (t), radius (r), and distance from center to external point (d) forms a right triangle where t² + r² = d² by the Pythagorean theorem. Given that the circle has radius r = 4 and the external point P is at distance d = 5 from center O, we form a right triangle with the radius as one leg, the tangent segment as the other leg, and OP as the hypotenuse, giving us t² + 4² = 5², so t² + 16 = 25, thus t² = 9 and t = √9 = 3. Choice A is correct because it applies the Pythagorean theorem correctly with specific values. Choice C incorrectly adds the radius and tangent length instead of using the Pythagorean theorem: d ≠ r + t, rather d² = r² + t². Key to tangent problems: remember that tangent ⊥ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. To find tangent length from external point, identify radius r and distance d to external point, then use t = √(d² - r²); if you recognize a Pythagorean triple, you can determine the answer immediately.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. The relationship between the tangent length (t), radius (r), and distance from center to external point (d) forms a right triangle where t² + r² = d² by the Pythagorean theorem. Given that the circle has radius r = 10 and the tangent length t = 24, we form a right triangle with the radius as one leg, the tangent segment as the other leg, and OP as the hypotenuse, giving us 24² + 10² = d², so 576 + 100 = d², thus d² = 676 and d = √676 = 26. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice A incorrectly reverses the Pythagorean relationship, treating the tangent length as the hypotenuse when actually the distance OP is the hypotenuse. Key to tangent problems: remember that tangent ⊥ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. To find tangent length from external point, identify radius r and distance d to external point, then use t = √(d² - r²); if you recognize a Pythagorean triple, you can determine the answer immediately.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. From a point outside a circle, exactly two tangent lines can be drawn to the circle, and the two tangent segments (from the external point to the points of tangency) have equal length. Because both tangent segments from P satisfy the same right triangle relationship (same distance d, same radius r), both have length t = √(d² - r²), making them equal; additionally, the configuration is symmetric about line OP. Choice C is correct because it correctly identifies equal lengths. Choice D incorrectly claims the two tangent segments have different lengths, but by symmetry and the Pythagorean theorem with the same parameters, they must be equal. From any external point, exactly two tangent lines can be drawn to a circle, and the two tangent segments are always equal in length due to the symmetry of the configuration. Remember: external point → 2 tangents; point on circle → 1 tangent; interior point → 0 tangents (no real tangent lines can be drawn).

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. The relationship between the tangent length (t), radius (r), and distance from center to external point (d) forms a right triangle where t² + r² = d² by the Pythagorean theorem. Given that the circle has radius r = 5 and the external point P is at distance d = 13 from center O, we form a right triangle with the radius as one leg, the tangent segment as the other leg, and OP as the hypotenuse, giving us t² + 5² = 13², so t² + 25 = 169, thus t² = 144 and t = √144 = 12. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice A incorrectly reverses the Pythagorean relationship, treating the tangent length as the hypotenuse when actually the distance OP is the hypotenuse. Key to tangent problems: remember that tangent ⊥ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. To find tangent length from external point, identify radius r and distance d to external point, then use t = √(d² - r²); if you recognize a Pythagorean triple, you can determine the answer immediately.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. A tangent line to a circle is a line that touches the circle at exactly one point, called the point of tangency, and is perpendicular to the radius at that point. The tangent line must be perpendicular to the radius OT at point T because this is the definition of tangency—any line through T that is not perpendicular to OT would either miss the circle entirely or intersect it at a second point. Choice A is correct because it correctly states the perpendicularity property. Choice B claims the tangent and radius are parallel or at an angle other than 90°, but the defining property of tangency is perpendicularity. Key to tangent problems: remember that tangent ⊥ radius at point of tangency creates a right triangle with the segment from center to external point as hypotenuse, allowing use of the Pythagorean theorem to find tangent length. When constructing tangents, the perpendicularity condition is essential: the tangent line must be perpendicular to the radius at the point where it touches the circle.

This question tests understanding of tangent lines from an external point to a circle and their geometric properties. From a point outside a circle, exactly two tangent lines can be drawn to the circle, and the two tangent segments (from the external point to the points of tangency) have equal length. From external point P, we can construct exactly two tangent lines because as we rotate a line through P, there are exactly two positions where the line is perpendicular to a radius, and those positions define the two tangent lines. Choice C is correct because it correctly counts tangents. Choice B incorrectly states there is only one tangent from an external point, but there are always exactly two tangent lines from any external point. From any external point, exactly two tangent lines can be drawn to a circle, and the two tangent segments are always equal in length due to the symmetry of the configuration. Remember: external point → 2 tangents; point on circle → 1 tangent; interior point → 0 tangents (no real tangent lines can be drawn).