Practice Constructing Tangents To Circles in Geometry with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
What this quiz covers
This quiz focuses on Constructing Tangents To Circles, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.
How to use this quiz
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
Question 1
A line p is tangent to circle ⊙Q at point J. Segment QJ is drawn, and the right angle between QJ and p at J is marked. Which reasoning correctly uses the radius–tangent relationship?
Because p is tangent at J, QJ⊥p at J.
Because is tangent at , .
Explanation: This question explores tangent properties in circle geometry. A tangent to a circle is a line that contacts the circle at precisely one point. This point is the point of tangency, labeled J. The radius QJ is perpendicular to the tangent p at J, forming the marked right angle. This reasoning correctly applies the radius-tangent perpendicularity theorem. A distractor like choice B incorrectly claims parallelism instead of perpendicularity. In solving, always connect the center to the tangent point and apply the perpendicular property.
Question 2
Circle ⊙O is shown with tangent line p touching the circle at point V. Radius OV is drawn, and the right angle at V is marked. Which property of tangents applies here?
A radius to the point of tangency is perpendicular to the tangent.
Question 3
Circle ⊙M has tangent line n touching it at point Q. The radius MQ is drawn, and the right angle between MQ and n at is marked. Which reasoning correctly uses the radius–tangent relationship?
Question 4
A tangent line r touches circle ⊙G at point H. The radius GH is drawn, and the right angle at H is marked. Which statement must be true at the point of tangency?
Line is perpendicular to radius at .
Question 5
Using compass and straightedge, a student constructs tangent lines from external point E to circle F by first drawing an auxiliary circle. The construction is successful, yielding two intersection points that determine the tangent lines. Which statement must be true about the auxiliary circle used in this construction?
The auxiliary circle has the same radius as the original circle and is centered at point E to ensure proper intersection angles
The auxiliary circle is concentric with the original circle but has radius equal to the distance to guarantee intersection
Question 6
During a compass and straightedge construction of tangent lines from external point P to circle O, a student draws the auxiliary circle with diameter PO but finds it doesn't intersect the original circle. What error did the student most likely make?
The compass opening was set incorrectly when drawing the original circle, making its radius too large for the construction
Point was actually chosen inside the original circle, making the auxiliary circle too small to reach the original circle
Question 7
A circle ⊙O is shown with tangent line ℓ touching the circle at point S. The radius is drawn, and the right angle at is marked between and . Which conclusion is NOT justified?
Question 8
Two circles have centers A and B respectively, with AB=10. Circle A has radius and circle has radius . If a common external tangent line is drawn to both circles, what is the distance between the points where this tangent touches each circle?
Question 9
Point W lies outside circle Z, and tangent segments WA and WB are drawn to the circle (with A and being points of tangency). If the radius of circle is and , what is the perimeter of quadrilateral ?
Question 10
A circle with center O is drawn. A line ℓ is tangent to the circle at T, and radius OT is drawn. The right angle between OT and ℓ is marked at . Which reasoning correctly uses the radius–tangent relationship?
Question 11
Line s is tangent to circle ⊙O at point U. Radius OU is drawn to the point of tangency, and the right angle at U is marked. Which angle relationship is guaranteed?
.
Question 12
A circle with center O is drawn. A tangent line ℓ touches the circle at S. Radius OS is drawn, and the right angle at S is marked. Which angle relationship is guaranteed?
.
Question 13
A point P is outside circle ⊙O. Two tangents from P touch the circle at points C and D. Radii OC and are drawn, and right angles are marked at and where each radius meets its tangent. Which statement must be true at the points of tangency?
Question 14
A tangent line t touches circle ⊙C at point R. The radius CR is drawn, and a right-angle marker at R shows CR. Which claim correctly describes the tangent?
Question 15
Circle ⊙G is shown with tangent line u touching the circle at point K. Radius GK is drawn, and the right angle at K between GK and is marked. Which angle relationship is guaranteed?
Question 16
A circle with center O is shown with a tangent line ℓ touching at A. Radius OA is drawn, and the right angle at A between OA and is marked. Which property of tangents applies here?
Question 17
A circle with center O is shown. Line ℓ is tangent to the circle at P, and radius OP is drawn. A right-angle marker at P shows the angle formed by OP and . Which angle relationship is guaranteed?
Question 18
A circle with center O is shown. Line ℓ is tangent to the circle at M, and radius OM is drawn. A right-angle marker at M indicates the angle between OM and . Which statement must be true at the point of tangency?
Question 19
A student attempts to construct a tangent line from external point R to circle C using compass and straightedge. The construction involves drawing a circle with diameter RC where C is the center of the original circle. Which statement best explains why this construction method works?
The intersection points lie on both circles, ensuring the connecting lines have equal slopes to the original circle's radius
Question 20
In the coordinate plane below, circle Q has center (0,4) and passes through (3,0). Point S is located at . How many distinct tangent lines can be drawn from point to circle ?
p
J
QJ∥p
Because QJ is a radius, line p meets the circle at two points.
Because QJ is a radius, point J must be the center.
A tangent is parallel to the diameter through the tangency point.
A tangent intersects the circle at exactly two points.
A tangent segment has both endpoints on the circle.
Explanation: This problem involves properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here labeled as V. At the point of tangency, the radius drawn from the center O to V is perpendicular to the tangent line p. This perpendicularity is a fundamental property of tangents, justifying its application. A common misconception is that tangents intersect at two points, but they touch at one. To solve similar problems, always find the radius to the tangent point and apply the perpendicular property.
Q
Since n touches the circle at Q, MQ must be perpendicular to n.
Since MQ is a radius, it must be parallel to tangent n.
Since n is a tangent, it must cross the circle at two points.
Since Q is on the circle, MQ must be a chord.
Explanation: This problem involves properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here labeled as Q. At the point of tangency, the radius drawn from the center M to Q is perpendicular to the tangent line n. This perpendicularity correctly uses the radius-tangent relationship, justifying the reasoning. A common misconception is that the radius is parallel to the tangent, but it is perpendicular. To solve similar problems, always find the radius to the tangent point and apply the perpendicular property.
r
GH
H
Line r intersects the circle again on the opposite side.
Point G lies on line r.
Segment GH is a tangent segment to the circle.
Explanation: This problem involves properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here labeled as H. At the point of tangency, the radius drawn from the center G to H is perpendicular to the tangent line r. This perpendicularity must be true at the point of tangency, justifying the statement. A common misconception is that the tangent intersects the circle again, but it does not. To solve similar problems, always find the radius to the tangent point and apply the perpendicular property.
EF
The auxiliary circle passes through both point E and center F, with its center located at the midpoint of segment EF
The auxiliary circle has center at point E and radius equal to the distance EF to create the proper geometric relationship
Explanation: When you encounter questions about compass and straightedge constructions for tangent lines from an external point, focus on the geometric properties that make the construction work. The key insight is understanding what auxiliary circle creates the right conditions for finding tangent points.The correct construction uses an auxiliary circle that passes through both the external point E and the center F of the original circle, with its center at the midpoint of segment EF. This creates a semicircle where EF is a diameter. When this auxiliary circle intersects the original circle, it produces two crucial intersection points. By the inscribed angle theorem, any angle inscribed in a semicircle is a right angle. This means the lines from these intersection points to point E are perpendicular to the radii of the original circle at those points—which is precisely the definition of a tangent line.Choice A is incorrect because having the same radius and centering at E doesn't create the perpendicular relationship needed for tangency. Choice B fails because being concentric (same center) with radius EF would place the auxiliary circle's center at F, not creating the necessary geometric configuration. Choice D places the center at E with radius EF, but this doesn't establish the right angle property required for tangent lines.Remember this pattern: geometric constructions often rely on creating specific angle relationships. When you see tangent line constructions, look for methods that guarantee perpendicularity between the tangent and radius.
P
The midpoint of segment PO was located incorrectly, causing the auxiliary circle to be centered at the wrong position
The auxiliary circle was drawn with PO as a chord rather than a diameter, resulting in a circle too small to intersect
Explanation: For the standard tangent construction to work, point P must be outside the original circle. The auxiliary circle has diameter PO (where O is the center of the original circle), so its radius is 2PO and it's centered at the midpoint of PO. If P is outside the original circle with radius r, then PO>r. The auxiliary circle extends from the midpoint toward O by distance 2PO, and since 2PO>1r= when PO>2r, it will intersect the original circle. However, if P is inside the original circle, then PO<r, making the auxiliary circle's radius 2PO<2r, and it cannot reach the original circle. Choice A is incorrect because the original circle's size doesn't affect intersection. Choice C is wrong because an incorrectly centered auxiliary circle would still likely intersect if properly sized. Choice D is incorrect because using PO as a chord (not diameter) would create a larger circle, not smaller.
OS
S
OS
ℓ
OS⊥ℓ at S.
Point S lies on the circle.
Line ℓ intersects the circle only at S.
Segment OS is a chord of the circle.
Explanation: This question asks which conclusion is NOT justified when dealing with tangents to circles. A tangent line touches a circle at exactly one point, and at this point of tangency, the tangent is perpendicular to the radius. Here, line ℓ is tangent to circle O at point S, which is the point of tangency. Since OS is a radius (connecting center O to point S on the circle), we can justify that OS is perpendicular to ℓ at S (choice A), that point S lies on the circle (choice B), and that line ℓ intersects the circle only at S (choice C). However, choice D claims that OS is a chord, which is incorrect—a chord connects two points on the circle, but OS connects the center to a point on the circle, making it a radius, not a chord. Students often confuse radii with chords; remember that all radii start at the center, while chords connect two points on the circle's circumference.
3
B
4
99
101
91
9
Explanation: For two external circles with centers distance d apart and radii r1 and r2, a common external tangent creates a trapezoid where the parallel sides are the radii to the tangent points. The distance between tangent points can be found using coordinate geometry or by recognizing that if we drop a perpendicular from one tangent point to the line through the other center parallel to the common tangent, we form a right triangle. The horizontal distance between centers is 10, and the vertical separation needed is ∣4−3∣=1 (difference in radii). Using the Pythagorean theorem on the right triangle formed: distance2=102−12=100−, so the distance is 99. Choice B (101) would result from incorrectly adding the radii: 102+12. Choice C (91) might come from an error like 102−32. Choice D (9) could result from simply subtracting: 10−1.
B
Z
5
WZ=13
WAZB
24
44
36
34
Explanation: When you see tangent segments drawn from an external point to a circle, think about the key properties: tangent segments from the same external point are equal in length, and each tangent is perpendicular to the radius at the point of tangency.Since WA and WB are tangent segments from point W to circle Z, we know WA=WB. To find these lengths, use the right triangles WAZ and WBZ. Each has a right angle where the tangent meets the radius (∠WAZ=∠WBZ=90°).In right triangle WAZ: WZ=13 (hypotenuse), AZ=5 (radius), so by the Pythagorean theorem: , which gives us , so and . Similarly, .The perimeter of quadrilateral WAZB is WA+AZ+ZB+BW=12+.Choice A (24) likely comes from adding only the tangent segments: 12+12=24, forgetting the two radii. Choice B (44) might result from incorrectly calculating the tangent length as 18 instead of 12, then adding all four sides. Choice C () could come from miscalculating the tangent segments as each, giving .Remember: tangent segments from an external point are always equal, and they form right angles with radii at the points of tangency—perfect setup for the Pythagorean theorem.
T
Since OT is a radius, OT must be parallel to the tangent ℓ.
Since ℓ touches the circle, it must pass through the center O.
Since ℓ is tangent at T, OT is perpendicular to ℓ at T.
Since OT meets ℓ at T, ℓ must cut the circle at two points.
Explanation: The skill involves understanding properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here point T. At the point of tangency, the radius to that point is perpendicular to the tangent line. Therefore, since ℓ is tangent at T, OT is perpendicular to ℓ at T, correctly using the relationship. A common misconception is that the tangent must pass through the center, but it does not. To solve similar problems, identify the radius to the point of tangency and apply the perpendicular property.
∠OUs=90∘
∠OUs=60∘.
∠UOs=90∘.
∠ between s and OU is 0∘.
Explanation: This problem involves properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here labeled as U. At the point of tangency, the radius drawn from the center O to U is perpendicular to the tangent line s. This perpendicularity guarantees a right angle at U, justifying the relationship. A common misconception is that the angle is acute or zero, but it is always right. To solve similar problems, always find the radius to the tangent point and apply the perpendicular property.
∠OSℓ=90∘
∠OSC=90∘ for any point C on the circle.
∠SOℓ=90∘.
∠OSS′=90∘ where S′ is any point on ℓ.
Explanation: The skill involves understanding properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here point S. At the point of tangency, the radius to that point is perpendicular to the tangent line. Therefore, the angle between OS and ℓ is a right angle, guaranteeing the relationship. A common misconception is that the right angle applies to any point on the circle, but it is specific to the tangency point. To solve similar problems, identify the radius to the point of tangency and apply the perpendicular property.
OD
C
D
OC and OD are chords of the circle.
Each tangent is perpendicular to its radius at the point of tangency.
Each tangent intersects the circle at two points.
The two tangents are parallel to each other.
Explanation: This question examines properties of tangent lines drawn from an external point to a circle. A tangent line touches a circle at exactly one point, and the fundamental property states that each tangent is perpendicular to the radius at its point of tangency. From external point P, two tangents are drawn touching the circle at points C and D. At each point of tangency, the radius (OC at point C, and OD at point D) is perpendicular to its respective tangent line, which is what choice B correctly states. Choice A incorrectly identifies radii OC and OD as chords—radii connect the center to points on the circle, while chords connect two points on the circle's circumference. Choice C wrongly claims tangents intersect at two points (they touch at exactly one). To work with tangents from external points, remember that each tangent maintains the perpendicular relationship with its radius at the point of tangency.
⊥
t
The tangent t passes through the center C.
The tangent t intersects the circle at exactly two points.
The tangent t is perpendicular to radius CR at R.
Segment CR is tangent to the circle at R.
Explanation: This problem involves properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here labeled as R. At the point of tangency, the radius drawn from the center C to R is perpendicular to the tangent line t. This perpendicularity correctly describes the tangent's relationship to the radius, justifying the claim. A common misconception is that the tangent passes through the center, but it does not. To solve similar problems, always find the radius to the tangent point and apply the perpendicular property.
u
∠GKu=180∘.
∠(GK,u)=90∘ at K.
∠GKu=45∘.
∠(GK,u) cannot be determined from the diagram.
Explanation: This problem evaluates tangent properties in circle geometry. A tangent to a circle is a line that meets the circle at precisely one point. This point is the point of tangency, labeled K. The radius GK is perpendicular to the tangent u at K, guaranteeing a right angle as marked. This relationship is ensured by the tangent-radius theorem. A misconception in choice A suggests a straight angle, ignoring the perpendicular property. In practice, find the radius to the tangent point to determine the right angle.
ℓ
A tangent is perpendicular to the radius at the point of tangency.
A tangent is parallel to the radius at the point of tangency.
A tangent always passes through the center of the circle.
A tangent intersects the circle at two points.
Explanation: The skill involves understanding properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here point A. At the point of tangency, the radius to that point is perpendicular to the tangent line. Therefore, the property that a tangent is perpendicular to the radius at the point of tangency applies here. A common misconception is that a tangent is parallel to the radius, but they are perpendicular. To solve similar problems, identify the radius to the point of tangency and apply the perpendicular property.
ℓ
∠(OP,ℓ)=90∘.
∠OPL=45∘.
Line ℓ is parallel to diameter through P.
Line ℓ crosses the circle again opposite P.
Explanation: The skill involves understanding properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here point P. At the point of tangency, the radius to that point is perpendicular to the tangent line. Therefore, the angle between OP and ℓ is a right angle, guaranteeing the relationship shown. A common misconception is that the tangent crosses the circle again, but it only touches at one point. To solve similar problems, identify the radius to the point of tangency and apply the perpendicular property.
ℓ
Point M is the center of the circle.
Line ℓ is perpendicular to OM at M.
Line ℓ intersects the circle at two points.
Segment OM is a tangent segment to the circle.
Explanation: The skill involves understanding properties of tangents to circles. A tangent to a circle is a line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of tangency, here point M. At the point of tangency, the radius to that point is perpendicular to the tangent line. Therefore, ℓ is perpendicular to OM at M, confirming the true statement. A common misconception is that the tangent intersects at two points, but it only touches at one. To solve similar problems, identify the radius to the point of tangency and apply the perpendicular property.
Any angle inscribed in a semicircle is a right angle, making the line from R perpendicular to the radius at the point of tangency
The two circles have the same center, guaranteeing that intersection points are equidistant from both circle centers
The diameter RC bisects the angle between the two tangent lines, creating symmetric intersection points on the auxiliary circle
Explanation: This construction works because of Thales' theorem: any angle inscribed in a semicircle is a right angle. When we draw a circle with diameter RC, any point T on this circle forms a right angle ∠RTC. If T is also on the original circle (at intersection points), then CT is a radius of the original circle, and ∠RTC=90° means RT is perpendicular to the radius CT. This is precisely the definition of a tangent line. Choice A is incorrect because tangent lines don't have 'equal slopes to the radius' - they're perpendicular to radii. Choice C is wrong because the circles have different centers (one centered at C, one centered at the midpoint of RC). Choice D is incorrect because the diameter doesn't bisect the angle between tangents.
(9,4)
S
Q
Zero tangent lines, since S lies inside the circle and no external tangents exist from interior points
Exactly one tangent line, since S lies on the circle and only one tangent exists at each point of tangency
Exactly two tangent lines, since S lies outside the circle and two external tangents can be drawn from any exterior point
Infinitely many tangent lines, since S lies on a diameter extension and tangents can be drawn at any angle through S
Explanation: First, I need to determine the position of point S relative to circle Q. The circle has center (0,4) and passes through (3,0), so its radius is (3−0)2+(0−4)2=. The distance from S(9,4) to center (0,4) is (9−0)2+(4−4)2. Since 9>5, point S lies outside the circle. From any external point, exactly two tangent lines can be drawn to a circle. Choice A is wrong because S is outside, not inside. Choice B is wrong because S is not on the circle (distance 9 ‚↠radius 5). Choice D is incorrect because even though S lies on a horizontal line through the center, this doesn't create infinitely many tangents - the number of tangents depends only on whether the point is inside, on, or outside the circle.