Constructing Tangents to Circles - Precalculus

Circle $O$ has center $O$ and radius $7$ . Point $P$ is an external point with $OP=25$ . A tangent from $P$ touches the circle at $T$ . Using the given circle and point, what is the length of the tangent segment $PT$ ?
Answer: $24$ 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 + 49 = 625, so t² = 576 and t = √576 = 24. This is a 7-24-25 Pythagorean triple, so we immediately recognize the tangent length is 24. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice A makes an arithmetic error, calculating √576 as 18 instead of 24. 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.
Circle $O$ has center $O$ and radius $8$ . An external point $P$ is located so that $OP=17$ . Tangent segment $PT$ touches the circle at $T$ . For the circle described, what is the length of the tangent segment $PT$ ?
Answer: $15$ 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.
A circle $O$ has center $O$ and radius $3$ . Point $P$ is outside the circle with $OP=5$ , and a tangent segment $PT$ touches the circle at $T$ . For the circle described, what is the angle between the radius $OT$ and the tangent line $PT$ at the point of tangency $T$ ?
Answer: $90^\circ$ This question tests understanding of tangent lines from an external point to a circle and their geometric properties. At the point of tangency, the radius and tangent line meet at a right angle (90°), which is the defining property of a tangent line. 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 C is correct because it correctly states the perpendicularity property of 90°. Choice B claims the angle is 60°, but the defining property of tangency is perpendicularity at 90°. 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.
A circle is tangent to both coordinate axes in the first quadrant. If a tangent line from point $(8, 0)$ to this circle has slope $m = -\frac{3}{4}$ , what is the radius of the circle?
Answer: $3$ Since the circle is tangent to both axes in the first quadrant, its center is at $(r,r)$ where $r$ is the radius. The tangent line from $(8,0)$ with slope $-\frac{3}{4}$ has equation $y = -\frac{3}{4}(x-8) = -\frac{3}{4}x + 6$ . The distance from center $(r,r)$ to this line must equal the radius: $\frac{|\frac{3}{4}r + r - 6|}{\sqrt{\frac{9}{16} + 1}} = r$ . This gives $\frac{|\frac{7}{4}r - 6|}{\frac{5}{4}} = r$ , so $|\frac{7}{4}r - 6| = \frac{5}{4}r$ . Since $r < 6$ , we have $6 - \frac{7}{4}r = \frac{5}{4}r$ , giving $6 = 3r$ , so $r = 3$ .
Circle $O$ has center $O(0,0)$ and radius $12$ . Point $P$ is an external point on the positive $x$ -axis, and a tangent segment $PT$ to the circle has length $9$ . For the circle described, what is the distance $OP$ ?
Answer: $\text{15}$ 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 = 12 and the tangent segment PT has length t = 9, we can find the distance OP using the Pythagorean theorem: 9² + 12² = d², so 81 + 144 = d², thus d² = 225 and d = √225 = 15. Choice B is correct because it applies the Pythagorean theorem correctly with the given values. Choice A incorrectly adds the radius and tangent length instead of using the Pythagorean theorem: d ≠ r + t = 21, rather d² = r² + t². To find the distance to an external point when given tangent length and radius, use d = √(t² + r²); this forms a 9-12-15 Pythagorean triple (a 3-4-5 triple scaled by 3).
A circle $O$ has center $O$ and radius $4$ . Point $P$ lies outside the circle with $OP=5$ . Tangent segments from $P$ touch the circle at points $T$ and $U$ . Using the given circle and point, what is the length of the tangent segment from $P$ to the circle (i.e., $PT$ )?
Answer: $3$ 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.
Circle $O$ has radius $4$ , and an external point $P$ satisfies $OP=5$ . Tangent segments from $P$ touch the circle at $T$ and $T'$ . Using the given circle and point, what is the length of the tangent segment $PT$ ?
Answer: $3$ 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.
Circle $O$ has center $O$ and radius $6$ . From an external point $P$ , a tangent segment $PT$ is drawn to the circle with $PT=8$ . If the tangent length is $8$ , what is the distance $OP$ from the center to the external point?
Answer: $10$ 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 = 6 and the tangent length t = 8, we can rearrange to find d = √(t² + r²) = √(64 + 36) = √100 = 10. This is a 6-8-10 Pythagorean triple (scaled 3-4-5 by 2), so we immediately recognize the distance is 10. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice C 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.
Circle $O$ has center $O$ and radius $5$ . Point $P$ is an external point with $OP=13$ . Two tangents from $P$ touch the circle at $T_1$ and $T_2$ . Using the given circle and point, how many tangent lines can be drawn from point $P$ to circle $O$ ?
Answer: $2$ 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 from an external point. 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. Remember: external point → 2 tangents; point on circle → 1 tangent; interior point → 0 tangents (no real tangent lines can be drawn).
Circle $O$ has center $O$ and radius $8$ . Point $P$ is an external point with $OP=17$ . Two tangent segments from $P$ touch the circle at points $T$ and $U$ . Using the given circle and point, what is the length of each tangent segment ( $PT$ and $PU$ )?
Answer: $15$ 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. 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 + 64 = 289, so t² = 225 and t = √225 = 15. This is a 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 D 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. 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.
Circle $O$ has center $O$ and radius $3$ . Point $P$ is outside the circle with $OP=5$ . A tangent from $P$ touches the circle at $T$ . Based on the circle described, what is the length of the tangent segment $PT$ ?
Answer: $4$ 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 = 3 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 + 9 = 25, so t² = 16 and t = √16 = 4. This is a 3-4-5 Pythagorean triple, so we immediately recognize the tangent length is 4. Choice B is correct because it applies the Pythagorean theorem correctly with specific values. Choice D uses the diameter instead of the radius in the calculation, which doubles a length and produces an incorrect result. 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.
A circle $\text{circle }O$ has center $O$ and radius $5$ . From an external point $P$ , a tangent line touches the circle at point $T$ . Based on the diagram description, what is the relationship between the tangent line $PT$ and the radius $OT$ at the point of tangency?
Answer: $PT \perp OT$ 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 B is correct because it correctly states the perpendicularity property. Choice A claims the tangent and radius are parallel, 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.
Circle $O$ has center $O$ and radius $5$ . Point $P$ is an external point (so $OP>5$ ). Using the given circle and point, how many tangent lines can be drawn from point $P$ to circle $O$ ?
Answer: $2$ 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).
A circle $O$ has center $O$ and radius $5$ . Point $P$ is outside the circle, and a tangent from $P$ touches the circle at $T$ . To construct a tangent from $P$ to circle $O$ , what must be true at the point of tangency $T$ ?
Answer: $OT \perp PT$ 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. 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. Remember: external point → 2 tangents; point on circle → 1 tangent; interior point → 0 tangents (no real tangent lines can be drawn).
Circle $O$ has radius $3$ , and an external point $P$ satisfies $OP=5$ . Tangent segments from $P$ touch the circle at points of tangency. Using the given circle and point, how many tangent lines can be drawn from point $P$ to circle $O$ ?
Answer: $2$ 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).
A circle $O$ has center $O$ and radius $5$ . An external point $P$ is such that $OP=13$ . From $P$ , a tangent segment $PT$ is drawn to the circle, touching the circle at $T$ . For the circle described, what is the length of the tangent segment $PT$ ?
Answer: $12$ 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.
Circle $O$ has center $O$ and radius $4$ . An external point $P$ satisfies $OP=10$ , and tangents from $P$ touch the circle at $T$ and $U$ . For the circle described, which of the following is true about the two tangent segments $PT$ and $PU$ ?
Answer: $PT = PU$ 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 their lengths cannot be compared without coordinates, 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. 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.
A circle $O$ has center $O$ . From an external point $P$ , a tangent line touches the circle at $T$ . Based on the diagram description, what is the relationship between the tangent line $PT$ and the radius $OT$ at the point of tangency $T$ ?
Answer: $PT$ is perpendicular to $OT$ . 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 B is correct because it correctly states the perpendicularity property. Choice A 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.
Circle $O$ has radius $6$ , and an external point $P$ satisfies $OP=10$ . Two tangents from $P$ touch the circle at $T$ and $T'$ . For the circle described, which of the following is true about the two tangent segments from $P$ ?
Answer: $PT = PT'$ 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 lengths cannot be compared without knowing ∠TPT', 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).
Circle $O$ has center $O$ and radius $4$ . Point $P$ is an external point with $OP=5$ . Two tangent segments from $P$ touch the circle at $T_1$ and $T_2$ . For the circle described, which of the following is true about the two tangent segments $PT_1$ and $PT_2$ ?
Answer: $PT_1 = PT_2$ 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 = 5, same radius r = 4), both have length t = √(d² - r²) = √(25 - 16) = √9 = 3, making them equal; additionally, the configuration is symmetric about line OP. Choice A is correct because it correctly identifies equal lengths. Choice D 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.

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