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Quizzes > Mathematics

Circle Relationships Mastery Test: Angles, Arcs, and Chords

Quick circle theorems practice quiz with instant results and helpful review links.

Editorial: Review CompletedCreated By: Gabi SzajdaUpdated Aug 28, 2025
Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art illustrating a trivia quiz for high school students to practice geometry skills with circle-related problems.

This circle relationships quiz helps you check your understanding of angles, arcs, chords, tangents, and secants. Work through 20 quick questions, see instant answers, and follow review notes as needed. For targeted practice, try the chords and arcs quiz or strengthen basics with the angle relationships quiz.

Which term describes a line segment that connects two points on a circle?
Secant
Tangent
Radius
Chord
A chord is a segment with both endpoints on the circle. Tangents touch the circle at one point while secants cut through it.
What is the longest chord in a circle called?
Arc
Radius
Diameter
Chord
The diameter is the longest chord because it passes through the center of the circle. It is exactly twice the length of the radius.
What is the formula for the circumference of a circle in terms of its radius r?
πr²
4Ï€r
Ï€r
2Ï€r
The circumference is calculated using the formula 2πr. Other formulas such as πr² are used for the area, not the circumference.
How does the measure of an inscribed angle compare to its intercepted arc?
It is twice the intercepted arc
It is half the measure of the intercepted arc
It is equal to the intercepted arc
It is the square root of the intercepted arc
According to the inscribed angle theorem, an inscribed angle measures half of its intercepted arc. This fundamental relationship is critical in circle geometry.
What is the measure of the angle between a radius and a tangent at the point of tangency?
45 degrees
90 degrees
0 degrees
180 degrees
A radius drawn to the point of tangency is perpendicular to the tangent, forming a 90-degree angle. This is a well-known property in circle geometry.
If a circle has a radius of 6 cm, what is its area?
12π cm²
36π cm²
24π cm²
18π cm²
The area of a circle is found using the formula A = πr². Substituting r = 6 gives A = π(6²) = 36π cm².
In a circle, if a central angle measures 60°, what is the measure of the intercepted arc?
90°
120°
60°
30°
A central angle has the same measure as its intercepted arc. Thus, a 60° central angle intercepts an arc measuring 60°.
When two chords intersect inside a circle, which relationship do they satisfy?
The sum of the segments of one chord equals the sum of the segments of the other chord
The product of the segments of one chord equals the product of the segments of the other chord
The segments are equal in length
They form congruent triangles
The intersecting chords theorem states that the product of the two segments of one chord is equal to the product of the two segments of the other chord. This property is useful in many circle geometry problems.
A tangent and a chord intersect at a point on a circle. What is the measure of the angle between them in relation to the intercepted arc?
It equals the intercepted arc
It is twice the intercepted arc
It equals half the intercepted arc
It is always 90° regardless of the arc
The tangent-chord angle theorem tells us that the angle between a tangent and a chord is half the measure of the intercepted arc. This principle is instrumental when solving problems involving tangents.
Which statement best describes the inscribed angle theorem?
An inscribed angle is always 90°
An inscribed angle is equal to the central angle that subtends the same arc
An inscribed angle measures half of its intercepted arc
An inscribed angle is supplementary to the intercepted arc
The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. This relationship is key in many circle geometry proofs.
If a circle has a radius of 10 cm and a central angle of 120°, what is the length of the intercepted arc?
(20Ï€/3) cm
(40Ï€/3) cm
(10Ï€/3) cm
20Ï€ cm
The arc length is calculated by (θ/360) Ã- 2Ï€r. Substituting θ = 120° and r = 10 cm gives (120/360) Ã- 20Ï€ = (1/3) Ã- 20Ï€ = (20Ï€/3) cm.
Why is a radius drawn to the midpoint of a chord perpendicular to that chord?
Because the chord is a tangent to the circle
Because the radius is parallel to the chord
Because the radius bisects the chord at a right angle
Because the chord always passes through the circle's center
A fundamental property of circles is that the perpendicular bisector of any chord will pass through the circle's center. Thus, a radius drawn to the midpoint of a chord is perpendicular to it.
For two secants intersecting outside a circle, the measure of the angle formed is given by which formula?
The difference of the intercepted arcs
The average of the intercepted arcs
½ (difference of the intercepted arcs)
½ (sum of the intercepted arcs)
When two secants intersect outside a circle, the angle between them is half the difference of the measures of the intercepted arcs. This theorem is a useful tool in solving circle problems that involve external intersections.
A circle has a circumference of 31.4 cm. What is its radius?
15 cm
5 cm
2.5 cm
10 cm
Using the circumference formula C = 2Ï€r, we solve for r: r = C/(2Ï€) = 31.4/(2Ï€) which is approximately 5 cm.
An arc of a circle measures 8 cm in length and the circle has a radius of 10 cm. What is the measure of the central angle in degrees?
Approximately 90°
Approximately 60°
Approximately 36°
Approximately 46°
The formula for arc length is (θ/360) Ã- 2Ï€r. Rearranging and solving for θ gives θ = (arc length Ã- 360)/(2Ï€r), which for 8 cm and r = 10 cm calculates to approximately 46°.
Two chords intersect inside a circle. If one chord is divided into segments of lengths 3 cm and 4 cm, and one segment of the other chord is 2 cm, what is the length of the unknown segment?
8 cm
7 cm
5 cm
6 cm
According to the intersecting chords theorem, the product of the segments of one chord equals the product of the segments of the other. Here, 3 Ã- 4 = 2 Ã- (unknown), so the unknown segment is 6 cm.
Two tangents are drawn from an external point to a circle and the angle between them is 40°. What is the measure of the intercepted arc between the points of tangency?
140°
40°
100°
220°
The angle between two tangents from an external point is equal to 180° minus the measure of the intercepted arc. Since the angle is 40°, the intercepted arc measures 180° âˆ' 40° = 140°.
The equation of a circle is given by x² + y² - 6x + 8y + 9 = 0. What is the radius of the circle?
5
3
6
4
Completing the square for both x and y terms, the equation can be rewritten to reveal the center and the square of the radius. The process shows that the radius squared is 16, so the radius is 4.
An inscribed quadrilateral has one angle measuring 70°. What must be the measure of its opposite angle?
70°
120°
110°
90°
In an inscribed quadrilateral, the opposite angles are supplementary. Since one angle is 70°, its opposite must be 180° âˆ' 70° = 110°.
If the angle between a tangent and a chord is 30°, what is the measure of the intercepted arc?
45°
30°
60°
90°
The tangent-chord angle theorem states that the angle between a tangent and a chord is half the measure of its intercepted arc. Therefore, with an angle of 30°, the intercepted arc must measure 60°.
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Study Outcomes

  1. Identify key components and properties of circles, including radii, chords, arcs, and tangents.
  2. Interpret and apply theorems related to central and inscribed angles in circle geometry.
  3. Analyze geometric diagrams to determine relationships between angles and segments in circles.
  4. Solve circle-related problems using algebraic methods and geometric reasoning.
  5. Evaluate and verify solutions to ensure accuracy in circle master challenges.

Circle Relationships Mastery Test Cheat Sheet

  1. All circles are similar - Because circles only differ in size but not in shape, mastering this concept makes scaling and comparing circles a breeze. Why guess? Similarity unlocks quick proportional reasoning when you need to jazz up your solutions. thecorestandards.org
  2. Equation of a circle - The formula (x - h)² + (y - k)² = r² is your magic spell for plotting circles on the coordinate plane. By tweaking h, k, and r, you can shift and grow your circle with mathematical flair. mathnirvana.com
  3. Area and circumference - Never forget A = πr² and C = 2πr when you need to measure a pizza or the boundary of a circle. These power-packed formulas let you calculate space and length faster than you can say "cheese." geeksforgeeks.org
  4. Central vs. inscribed angles - A central angle is always twice an inscribed angle that spans the same arc - no ifs, ands, or buts! This golden rule helps you crack angle-chasing puzzles in circles. brilliant.org
  5. Thales's theorem - Any angle inscribed in a semicircle is a perfect right angle, thanks to Thales's wisdom. Use this to spot or prove right angles in complex circle diagrams like a pro. en.wikipedia.org
  6. Perpendicular bisector of a chord - The line that cuts a chord in half at 90° always runs through the circle's center. This trick is gold when you're hunting for that elusive center point. brilliant.org
  7. Radius-tangent perpendicularity - The radius hitting the tangent line at the precise tangency point is always 90°, creating a neat right angle. This property is your go-to for tackling tangent proofs. thecorestandards.org
  8. Equal chords subtend equal angles - When two chords are the same length, they carve out equal central angles - no surprises here! This rule is great for crafting super-symmetric circle proofs. byjus.com
  9. Cyclic quadrilateral angle sum - Opposite angles in a cyclic quadrilateral add up to 180°, making angle-hunting more systematic than random guesswork. It's a must-know for circle-based quadrilateral problems. brilliant.org
  10. Nine-point circle - This magical circle zips through nine key points of a triangle: midpoints of sides, feet of altitudes, and more. It's a high-level gem for advanced geometric constructions and olympiad problems. en.wikipedia.org
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