12 multiple-choice questions, progressively harder.
PA and PB are tangent to the same circle from one strict external point. Their lengths are 2y+72y+72y+7 and 5y−115y-115y−11. Which pair gives y and the common length?
y=6y=6y=6
Length 121212
y=3y=3y=3
Length 191919
Solution
Correct answer: D
Equalize the two lengths.
Substitution gives the common positive length.
A circle has radius 151515 cm. PT is tangent at T with length 888 cm. The segment OP meets the circle at A between O and P. Find AP.
Correct answer: B
Use the right triangle at T to find OP.
Since A lies between O and P, subtract radius OA.
A circle has radius 333 units. Its center O is 58\sqrt{58}58 units from strict external point P. PT is tangent at T. Find PT.
Correct answer: C
The contact radius is perpendicular to PT.
Two circles have radii 131313 and 555 units and centers 888 units apart. Which description of their common tangents is correct?
Correct answer: A
The center distance equals the positive radius difference. The circles touch internally. Their unique shared tangent has both centers on the same side, so it is external.
Circles of radii 777 and 444 units first have centers 121212 units apart. The center distance then becomes 111111 units. What are the total common-tangent counts, in that order?
The radius sum is 111111. At distance 121212, the circles are separate and have four common tangents. At distance 111111, they touch externally and the two internal tangents merge into one. There are then three.
Two circles have radii 666 and 222 units. Their center distance is 555 units. Which common-tangent count is correct?
The radius difference is 444 and the radius sum is 888. Center distance 555 lies strictly between these boundaries, so the circles intersect. Only the two external common tangents exist.
T is on a circle of radius r centered at O. P is on the tangent at T, and P differs from T. What does the right triangle OTP show?
OT is perpendicular to PT. Since P differs from T, PT is positive.
Thus P is a strict external point.
Which information is sufficient to apply the equal-tangent-segments theorem to segments PA and PB?
The theorem needs both tangencies to the same circle, a shared starting point, and that point strictly outside. Merely ending on a circle makes secant or chord segments possible. Different circles supply no shared radius leg.
Two circles both have radius 444 units. Their centers are distinct and 666 units apart. How many common tangent lines do they have?
The radius difference is zero, while the sum is 888. Center distance 666 is between them, so the circles intersect twice and have two common external tangents. Equal radii do not make circles coincident when their centers differ.
Two circles have radii 111111 and 333 units. Which center distance gives exactly one common tangent line?
Exactly one common tangent occurs at internal contact.
Distance zero gives unequal concentric circles and no common tangent. Distance 141414 gives external contact and three; distance 161616 gives four.
Circle C and circle D have the same center and each has radius 777 cm. Which statement is true?
The center and radius determine the circle, so C and D coincide. Every tangent to C is also tangent to D, giving infinitely many common tangents.
Two circles are internally tangent at T. Their radii are 111111 cm and 444 cm. What are the center distance and total number of common tangents?
Distance 777 cm
333 common tangents
111 common tangent
Distance 151515 cm
Internal contact uses the radius difference.
The common tangent at T is external because the two centers lie on its same side. It is the only common tangent.
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