12 multiple-choice questions, progressively harder.
A circle has radius 121212 cm. PT is tangent at T and OP has length 131313 cm. Find PT.
Solution
Correct answer: C
The right angle is at T.
A circle has radius 121212 units. A tangent segment from P to its contact T has length 999 units. How far is P from center O?
Correct answer: A
The contact radius and tangent segment are perpendicular legs.
Point TTT is on a circle centered at OOO with radius 101010 cm. Point XXX differs from TTT, with TX=24TX=24TX=24 cm and OX=26OX=26OX=26 cm. What follows?
Correct answer: D
The given lengths satisfy the Pythagorean converse.
The right angle is between OT and TX. Since T is on the circle, the radius-tangent converse proves TX tangent.
A circle has radius 666 units. A line has perpendicular center distance 444 units. What is its number of circle intersections?
Correct answer: B
The perpendicular distance is smaller than the radius. There is one intersection on each side of the foot.
PA and PB are tangent segments from one strict external point to one circle. Their lengths are 4x−34x-34x−3 and x+12x+12x+12. Find x.
Set the two tangent lengths equal.
Both lengths are then 171717, so the lengths are positive.
Two circles have radii 101010 cm and 666 cm. Their centers are 161616 cm apart. How many common tangent lines do they have?
The center distance equals the radius sum, so the circles touch externally. The common internal tangent is one line at the shared contact. The two common external tangents remain.
Two circles have radii 888 and 555 units, with center distance 777 units. Which common tangents exist?
The radius difference is 333 and the sum is 131313. The center distance lies strictly between them, so the circles intersect twice. They have two common external tangents and no internal tangents.
Two circles have radii 121212 and 777 units, with centers 333 units apart. How many common tangent lines do they have?
The radius difference is 555, greater than center distance 333. The smaller circle lies strictly inside the larger without contact. No line can touch both without crossing the larger.
A common tangent places the two circle centers on opposite sides of its line. What is it called?
Opposite-side centers are the definition of a common internal tangent. A common chord has two endpoints on both circles and is not this tangent line.
Two circles have radii 999 cm and 222 cm. Their centers are 777 cm apart. How many distinct common tangent lines exist?
The center distance equals the positive radius difference. The circles are internally tangent. Their common external tangent at the contact is the only common tangent line.
In the radius-tangent proof, suppose the perpendicular foot H differs from contact T. Reflect T across the perpendicular through the center. Why does this contradict tangency?
Call the reflected point U. Reflection fixes the center and preserves distance, so U remains on the circle. It also stays on the original line. Since H differs from T, the image differs from T, producing two intersections instead of one.
A circle has radius 555 units. Point P is 333 units from its center. Can P be the starting point of two positive tangent contact segments?
The center distance is smaller than the radius, so P is inside. No tangent line contains an interior point. There are no tangent contact segments from P.
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