12 multiple-choice questions, progressively harder.
A line meets a circle at two distinct points A and B. What is the line called?
Solution
Correct answer: B
A secant is defined by two distinct circle intersections. A tangent has exactly one. A radius is a segment from the center to the circle, not the entire line.
A circle has radius 111111 units. The perpendicular distance from its center to line l is 111111 units. How many intersections does l have with the circle?
Correct answer: D
The closest point of the line is exactly one radius from the center. Every other line point is farther away.
Line PT is tangent to a circle centered at O at point T. Which segment is perpendicular to PT?
Correct answer: A
The radius-tangent theorem uses the radius ending at the contact point. Here that is OT. The theorem gives no perpendicularity with an arbitrary chord or radius.
A circle has center O and radius r. Which condition makes P a strict external point?
Correct answer: C
A strict external point is farther from the center than every circle point. Its center distance must exceed the radius.
A circle has radius 777 cm. Point P is 252525 cm from its center O. PT is tangent at T. Find the length of segment PT.
The contact radius is perpendicular to PT, so OP is the hypotenuse.
Segments PA and PB are tangent to the same circle from the same strict external point P. If PA has length 141414 cm, what is PB?
The two segments share a strict external starting point and touch the same circle. Equal tangent segments therefore have the same length.
Point T lies on a positive-radius circle. How many tangent lines through T does the circle have?
Any tangent through T must be perpendicular to the radius at T. There is exactly one perpendicular line there, and the converse theorem proves it tangent.
Point P is strictly inside a circle. How many tangent lines to the circle pass through P?
Every tangent line has perpendicular distance from the center equal to the radius. An interior point is closer to the center, so it cannot lie on such a line.
Two circles have radii 444 cm and 333 cm. They touch externally. What is their center distance?
For external contact, the contact point lies between the centers. Add the two radii.
A line is tangent to two circles. Their centers lie on the same side of that line. How is this common tangent classified?
The definition uses the centers’ sides of the tangent line. Centers on the same side make it a common external tangent.
Two circles have radii 555 cm and 222 cm, with centers 101010 cm apart. How many distinct common tangent lines do they have?
The radius sum is 777 cm, less than the center distance. The circles are separate, with neither inside the other. There are two external and two internal tangents.
Two circles share a center, but their positive radii differ. How many common tangent lines do they have?
A tangent line must be one radius from the center. A single line has only one perpendicular distance from that shared center, so it cannot equal two different radii.
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