Constructing Tangents to Circles
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Geometry › Constructing Tangents to Circles
A line $p$ is tangent to circle $\odot Q$ 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 $J$, $QJ \parallel p$.
Because $QJ$ is a radius, point $J$ must be the center.
Because $QJ$ is a radius, line $p$ meets the circle at two points.
Because $p$ is tangent at $J$, $QJ \perp p$ at $J$.
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.
Line $s$ is tangent to circle $\odot O$ 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?
$\angle OUs=90^\circ$.
$\angle$ between $s$ and $OU$ is $0^\circ$.
$\angle OUs=60^\circ$.
$\angle UOs=90^\circ$.
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.
Circle $\odot M$ has tangent line $n$ touching it at point Q. The radius $MQ$ is drawn, and the right angle between $MQ$ and $n$ at $Q$ is marked. Which reasoning correctly uses the radius–tangent relationship?
Since $Q$ is on the circle, $MQ$ must be a chord.
Since $n$ is a tangent, it must cross the circle at two points.
Since $n$ touches the circle at $Q$, $MQ$ must be perpendicular to $n$.
Since $MQ$ is a radius, it must be parallel to tangent $n$.
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.
A tangent line $r$ touches circle $\odot G$ 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 $r$ intersects the circle again on the opposite side.
Line $r$ is perpendicular to radius $GH$ at $H$.
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.
A circle with center $O$ is drawn. A line $\ell$ is tangent to the circle at T, and radius $OT$ is drawn. The right angle between $OT$ and $\ell$ is marked at $T$. Which reasoning correctly uses the radius–tangent relationship?
Since $OT$ meets $\ell$ at $T$, $\ell$ must cut the circle at two points.
Since $\ell$ touches the circle, it must pass through the center $O$.
Since $\ell$ is tangent at $T$, $OT$ is perpendicular to $\ell$ at $T$.
Since $OT$ is a radius, $OT$ must be parallel to the tangent $\ell$.
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.
A circle with center $O$ is drawn. A tangent line $\ell$ touches the circle at S. Radius $OS$ is drawn, and the right angle at $S$ is marked. Which angle relationship is guaranteed?
$\angle SO\ell = 90^\circ$.
$\angle O S C = 90^\circ$ for any point $C$ on the circle.
$\angle OSS' = 90^\circ$ where $S'$ is any point on $\ell$.
$\angle OS\ell = 90^\circ$.
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.
A point $P$ is outside circle $\odot O$. Two tangents from $P$ touch the circle at points C and D. Radii $\overline{OC}$ and $\overline{OD}$ are drawn, and right angles are marked at $C$ and $D$ where each radius meets its tangent. Which statement must be true at the points of tangency?
$\overline{OC}$ and $\overline{OD}$ are chords of the circle.
Each tangent is perpendicular to its radius at the point of tangency.
The two tangents are parallel to each other.
Each tangent intersects the circle at two points.
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.
A circle $\odot O$ is shown with tangent line $\overleftrightarrow{\ell}$ touching the circle at point S. The radius $\overline{OS}$ is drawn, and the right angle at $S$ is marked between $\overline{OS}$ and $\ell$. Which conclusion is NOT justified?
$\overline{OS} \perp \overleftrightarrow{\ell}$ at $S$.
Line $\ell$ intersects the circle only at $S$.
Segment $\overline{OS}$ is a chord of the circle.
Point $S$ lies on 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.
A tangent line $t$ touches circle $\odot C$ at point R. The radius $CR$ is drawn, and a right-angle marker at $R$ shows $CR \perp t$. Which claim correctly describes the tangent?
The tangent $t$ intersects the circle at exactly two points.
The tangent $t$ passes through the center $C$.
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.
Circle $\odot G$ 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 $u$ is marked. Which angle relationship is guaranteed?
$\angle G K u=180^\circ$.
$\angle(GK,u)=90^\circ$ at $K$.
$\angle G K u=45^\circ$.
$\angle(GK,u)$ 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.