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Quizzes > Mathematics

Point of Tangency Quiz: Spot Tangent Lines in Circles

20 quick questions for circle tangents practice. Instant results.

Editorial: Review CompletedCreated By: Rodmar PulisUpdated Aug 27, 2025
Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting Tangent or Not high school geometry quiz.

This point of tangency quiz helps you tell if a line touches a circle at one point or cuts through it. Answer 20 quick questions to practice spotting tangents and secants, with instant results and simple diagrams. After you finish, try our angles in circles quiz and the chords and arcs quiz to deepen your circle skills.

Which of the following best defines a tangent to a circle?
A line that cuts the circle at two points.
A line that touches the circle at exactly one point.
A line that does not touch the circle.
A line that lies entirely inside the circle.
A tangent touches a circle at exactly one point and does not intersect it further. This is the defining property of a tangent line.
At the point of tangency, a tangent line to a circle is perpendicular to which of the following?
The diameter of the circle.
The radius drawn to the point of tangency.
Any secant passing through the circle.
Any chord of the circle.
The tangent line is perpendicular to the radius at the point of contact. This is a fundamental property of circles.
If a line touches a circle at one and only one point, it is considered a tangent line. Is this statement true or false?
False
Only if it is perpendicular to the radius
True
Depends on the circle's size
A line that touches a circle at exactly one point is, by definition, a tangent line. The other conditions are either unnecessary or incorrect.
Which statement about a tangent line is incorrect?
It intersects the circle in two distinct points.
It lies outside the circle except at the point of contact.
It touches the circle at only one point.
It is perpendicular to the radius at the point of tangency.
A tangent line should only touch a circle at one point; if it intersects the circle in two points, it is a secant. Thus, Option B is the incorrect statement.
If a line crosses a circle and produces two intersection points, what is the name of that line?
Secant
Radius
Tangent
Chord
A line that crosses a circle and intersects it at two points is called a secant line, not a tangent. This is an important distinction in circle geometry.
Given a circle with center O, a line touches the circle at point T. Which angle is 90° in this configuration?
The angle between a chord and the tangent.
The angle at the center subtended by the tangent.
The angle between two intersecting tangents.
The angle between the tangent line and the radius OT.
According to circle geometry, the tangent line is perpendicular to the radius at the point of tangency. This creates a 90° angle between them.
If a perpendicular is drawn from the center of a circle to a tangent line, where does it meet the tangent?
It never meets the tangent line.
At the point of tangency.
At the midpoint of the tangent segment outside the circle.
At a point not on the circle.
By definition, the perpendicular from the center to a tangent touches the tangent exactly at the point of tangency. This is a key characteristic of tangents.
What is the distance from the center of a circle to a tangent line in relation to the circle's radius?
Unrelated to the radius.
Greater than the radius.
Less than the radius.
Equal to the radius.
The distance from the center of a circle to its tangent line is exactly equal to the radius. This distance criterion is used to determine tangency.
When two tangents are drawn from an external point to a circle, what is true about their lengths?
They are equal in length.
Their lengths depend on the circle's diameter.
The tangent closer to the center is shorter.
They have different lengths.
A fundamental property of tangents drawn from an external point is that they are congruent. This equality is often used in problem-solving.
Which theorem confirms that the tangent to a circle is perpendicular to its radius at the point of contact?
Alternate Interior Angles Theorem
Tangent-Radius Theorem
Inscribed Angle Theorem
Pythagorean Theorem
The Tangent-Radius Theorem states that a tangent line is perpendicular to the radius at the point where it touches the circle. This theorem is a foundational concept in circle geometry.
How can you verify if a given line is tangent to a circle using distance measurement?
Compare the distance from the center to the line with the circle's radius.
Determine the circle's diameter.
Check the length of the line inside the circle.
Measure the distance between the intersections of the line with the circle.
A line is tangent to a circle if the distance from the circle's center to the line equals the radius. This method is frequently used in analytical geometry.
For a circle with center (h, k) and radius r, and a line given by ax + by + c = 0, which condition must hold true for the line to be tangent to the circle?
|ah + bk + c| > r√(a² + b²)
|ah + bk + c| < r√(a² + b²)
|ah + bk + c| = r√(a² + b²)
√(a² + b²) = r
The distance from the center (h, k) to the line ax + by + c = 0 is given by |ah + bk + c| / √(a² + b²). For tangency, this distance must equal r, leading to the given condition.
Which of the following cannot be a tangent to a circle with center O?
A line that is perpendicular to a radius.
A line that touches the circle at one point.
A line that grazes the circle externally.
A line that passes through O.
A tangent line cannot pass through the center because it would intersect the circle at more than one point. Therefore, any line passing through the center cannot be a tangent.
When the system of equations representing a circle and a line yields a discriminant of zero, what does this imply about the line?
The circle is degenerate.
The line is a secant intersecting the circle at two points.
The line does not intersect the circle.
The line is tangent to the circle.
A discriminant of zero indicates a single intersection point between the line and the circle. This scenario corresponds to the line being tangent to the circle.
According to the Tangent-Chord Angle Theorem, the angle between a tangent and a chord is equal to which of the following?
A quarter of the measure of the intercepted arc.
Half the measure of the intercepted arc.
The measure of the intercepted arc.
Twice the measure of the intercepted arc.
The Tangent-Chord Angle Theorem states that the angle formed between a tangent and a chord at the point of tangency is half the measure of the intercepted arc. This theorem helps in solving various circle geometry problems.
Two tangents are drawn from an external point P to a circle, touching the circle at points T and Q. If ∠TPQ measures 80°, what is the measure of the intercepted arc TQ?
100°
140°
80°
160°
The angle between two tangents from an external point equals 180° minus the intercepted arc between the tangency points. Hence, if ∠TPQ is 80°, the intercepted arc TQ measures 100°.
The Alternate Segment Theorem in circle geometry states that the angle between a tangent and a chord through the point of tangency is equal to what?
The angle in the alternate segment of the circle.
The corresponding central angle.
The inscribed angle on the same side of the chord.
Double the measure of the intercepted arc.
According to the Alternate Segment Theorem, the angle between the tangent and the chord is equal to the angle in the alternate segment made by the chord. This is a crucial result in problems involving tangents and circles.
For a circle with radius r, which equation must a line satisfy to be tangent to the circle?
It must lie at a distance equal to r from the center.
It must lie at a distance less than r from the center.
It can lie at any distance since tangency is defined by intersection points.
It must lie at a distance greater than r from the center.
A tangent line touches a circle at exactly one point, meaning the perpendicular distance from the center to the line must equal the radius, satisfying the condition for tangency.
Given a circle with center (3, -2) and radius 5, if c is positive and the line 4x - 3y + c = 0 is tangent to the circle, what is the value of c?
10
-43
7
0
The distance from the circle's center to the line is |4(3) - 3(-2) + c|/√(4² + (-3)²) = |18+c|/5. Setting it equal to 5 gives |18+c| = 25. With c positive, 18+c = 25, so c = 7.
A line l is tangent to two circles with centers O1 and O2 at points T1 and T2, respectively. What must be true about the radii drawn to T1 and T2?
They are both perpendicular to the tangent line l.
They are parallel to each other.
They intersect at the midpoint of the tangent segment.
They are collinear with the points of tangency.
It is a key property of tangents that the radius drawn to the point of tangency is perpendicular to the tangent line. Therefore, both radii from O1 to T1 and from O2 to T2 are perpendicular to line l.
0
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Study Outcomes

  1. Analyze geometric figures to determine whether a line is tangent to a circle.
  2. Identify the key properties that differentiate a tangent from other lines.
  3. Apply the perpendicularity criterion between a radius and a tangent at the point of contact.
  4. Evaluate diagrammatic representations to confirm the presence or absence of a tangent line.
  5. Synthesize theoretical knowledge and practical examples to make informed decisions on tangent-related problems.

Tangent Line Quiz: Determine if It's Shown Cheat Sheet

  1. Tangent line definition - A tangent line touches a curve at exactly one point, matching the curve's slope right there. It's like the curve's BFF that shares its exact direction at that spot.
  2. Tangent line equation - The magic formula y − f(x₀) = f′(x₀)(x − x₀) uses the derivative at x₀ and a known point on the curve. Plug in f(x₀) and f′(x₀) to reveal exactly how the line hugs the curve.
  3. Circle tangents - For a circle, a tangent intersects at exactly one point and stands perfectly perpendicular to the radius there. It's nature's way of ensuring a perfect "kiss" between line and circle.
  4. Point of tangency - This is the exact coordinate where the curve and its tangent meet, sharing both position and slope. Identifying this point is crucial for drawing or calculating the tangent line.
  5. Finding a tangent - Calculate f(a) and f′(a) for your function at x = a, then plug into y − f(a) = f′(a)(x − a). Voilà - your live-action tangent is born!
  6. Secant vs. tangent - A secant line cuts through a curve at two or more points, while a tangent just kisses it once. As those two points on a secant get infinitely close, the secant morphs into your tangent.
  7. Linear approximation - Tangent lines let you approximate a function's value near a point by treating the curve like a straight line. It's a speedy shortcut for estimates without heavy calculation.
  8. Derivative as slope - The derivative at a point is literally the slope of the tangent line there. Mastering derivatives means mastering the direction in which your curve's heading.
  9. Best local fit - The tangent line gives the best straight-line approximation of the curve near a chosen point. Zoom in close enough and the curve and its tangent become practically indistinguishable.
  10. Applications in rates of change - From velocity in physics to optimization in economics, tangent lines and derivatives are everywhere. Nail these concepts and you'll be unstoppable in solving real-world problems!
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