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Quizzes > High School Quizzes > Mathematics

Chords and Arcs Quick Check Practice Quiz

Test your skills with engaging practice questions

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art depicting trivia on Arc and Chord Blitz challenges for high school students.

Which of the following best defines an arc in a circle?
An angle inside the circle
The diameter
The curved part of the circle's circumference
A straight line through the circle
An arc is the curved portion of a circle's circumference. It does not refer to any straight line, diameter, or internal angle.
What does a chord in a circle represent?
A ray starting from the center
The longest distance through the circle
A line outside of the circle
A line segment with both endpoints on the circle
A chord is defined as a straight line connecting two points on the circle's circumference. It is distinct from a ray or the diameter, although a diameter is a special type of chord.
If a chord is the same length as the radius, what special triangle is formed when you connect the chord's endpoints with the circle's center?
Right triangle
Obtuse triangle
Scalene triangle
Equilateral triangle
When the chord equals the radius in length, the two radii and the chord form a triangle with all equal sides. This results in an equilateral triangle, where every angle measures 60°.
In a circle, which of the following is always true about a diameter?
It is parallel to all chords
It is the shortest chord in the circle
It is the longest chord in the circle
It does not pass through the center
A diameter passes through the center of the circle and represents the longest possible chord. The other options do not accurately describe the properties of a diameter.
What is the measure of a full circle in degrees?
100°
360°
180°
90°
A full circle is measured as 360°, which is the standard basis for determining portions of a circle such as arcs and sectors. This foundational fact is essential for working with circle geometry.
Which formula is used to calculate the length of an arc (s) given the radius (r) and the central angle in radians (θ)?
s = (1/2)rθ
s = r²θ
s = rθ
s = 2πrθ
The correct formula to calculate arc length is s = rθ when the angle is provided in radians. The other formulas include incorrect coefficients or operations.
If two chords in a circle are congruent, what can be said about the arcs they subtend?
They subtend congruent arcs
They subtend arcs that are complementary
They subtend arcs that are supplementary
They subtend arcs with different lengths
In circle geometry, congruent chords always subtend arcs of equal measure due to the inherent symmetry of the circle. This relationship is a fundamental property of circles.
Given a circle with radius 10 units, what is the length of an arc corresponding to a 90° central angle?
31.42 units
20 units
Approximately 15.71 units
25 units
Convert 90° to radians, which gives π/2. Using the formula s = rθ, the arc length is 10 à - (π/2), which is approximately 15.71 units. The other options result from miscalculations.
Which of the following statements is true about inscribed angles in a circle?
An inscribed angle is twice the measure of its intercepted arc
An inscribed angle is equal to its intercepted arc
An inscribed angle and its intercepted arc sum to 90°
An inscribed angle is half the measure of its intercepted arc
The inscribed angle theorem clearly states that an inscribed angle is half the measure of its intercepted arc. This property is crucial for solving many circle geometry problems.
What is the relationship between a chord's distance from the center of a circle and its length?
Only the radius determines the chord's length
Distance from the center does not affect the chord's length
A chord closer to the center is longer
A chord closer to the center is shorter
Chords located closer to the center of a circle tend to be longer because they are subtended by larger central angles. This geometric relationship is often proved using perpendicular bisector properties.
An arc's measure can be described as the fraction of 360° that the arc covers. If an arc spans 45°, what fraction of the total circumference does it represent?
1/4
1/8
1/10
1/6
Since 45° is 1/8 of 360°, the arc covers one-eighth of the full circumference. This fraction is found by dividing the arc measure by the total 360°.
When two chords intersect inside a circle, which of the following is true?
The product of the segments of one chord equals the product of the segments of the other chord
The chords are congruent
The chords are perpendicular
The sums of the segments of the chords are equal
The intersecting chords theorem states that the product of the segments of one chord is equal to the product of the segments of the other chord. This result is widely used in solving problems involving intersecting chords.
What does the perpendicular bisector of a chord in a circle always pass through?
The center of the circle
No fixed point
The circle's circumference
The midpoint of the arc
A fundamental property of a circle is that the perpendicular bisector of any chord passes through the center. This fact is frequently used to locate the center of the circle.
If the measure of an inscribed angle is 30°, what is the measure of its intercepted arc?
90°
60°
120°
30°
According to the inscribed angle theorem, the intercepted arc is twice the measure of the inscribed angle. Therefore, an inscribed angle of 30° intercepts a 60° arc.
How do you find the central angle in radians if you know the arc length (s) and the radius (r)?
θ = s*(π/r)
θ = r/s
θ = s/r
θ = r - s
Rearranging the arc length formula s = rθ gives θ = s/r, which is used to determine the central angle in radians. The other options do not correctly isolate θ.
In circle geometry, if two arcs are congruent, which of the following can be inferred about the corresponding chords, assuming they lie in the same circle?
One chord is longer than the other
The chords are perpendicular
The chords are congruent
The chords subtend different central angles
In a circle, congruent arcs subtend chords that are equal in length. This relationship is a consequence of the circle's symmetry.
A circle has a radius of 12 units. A chord is at a distance of 5 units from the center. Using the Pythagorean theorem, what is half the length of this chord?
√(144 - 5)
√(12² - 5²) = √119
√(144 + 25)
√(144/5)
A right triangle can be formed by the radius, the distance from the center to the chord, and half the chord length. Using the Pythagorean theorem, half the chord length is √(r² - d²) = √(144 - 25) = √119.
Determine the length of a chord if the central angle is 2 radians and the radius is 7 units, using the chord length formula.
14 Ã - cos(1)
7 Ã - tan(1)
7 Ã - sin(2)
2 Ã - 7 Ã - sin(1) = 14 sin(1)
The chord length can be calculated using the formula 2r sin(θ/2). With r = 7 and θ = 2 radians, this gives 2 à - 7 à - sin(1), which simplifies to 14 sin(1). The other options do not apply the formula correctly.
A circle has a chord that subtends a central angle of 1.2 radians. Calculate the chord length for a circle with radius 5 units.
5 Ã - sin(1.2)
10 Ã - cos(0.6)
2 Ã - 5 Ã - sin(0.6) = 10 sin(0.6)
2 Ã - 5 Ã - cos(0.6)
Using the chord length formula 2r sin(θ/2) with r = 5 and θ = 1.2 radians, we obtain 2 à - 5 à - sin(0.6), which simplifies to 10 sin(0.6). Incorrect formulas were applied in the other options.
Consider two intersecting chords in a circle where one chord is divided into segments of lengths 3 and x, and the other is divided into segments of lengths 2 and 6. What is the value of x using the intersecting chords theorem?
x = 3
x = 12
x = 4
x = 6
The intersecting chords theorem states that the product of the segments of one chord equals the product of the segments of the other. Here, 3 Ã - x = 2 Ã - 6, which means 3x = 12 and therefore x = 4.
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Study Outcomes

  1. Analyze the relationship between arcs and chords in circles.
  2. Apply geometric formulas to calculate arc measures and chord lengths.
  3. Solve problems involving angle and chord properties within circles.
  4. Evaluate solutions using circle geometry principles.

Chords & Arcs Quick Check Cheat Sheet

  1. Understanding Chords - A chord is a line segment that connects any two points on a circle's edge. The biggest chord you can draw is the diameter, which slices right through the center and divides the circle into two perfect halves. Imagine these chords as secret passageways across your circular arena! Byju's: Chord of a Circle
  2. Chord Length Formula - To calculate how long a chord is, use the magic formula: 2 × √(r² − d²), where r is the circle's radius and d is the shortest distance from the chord to the center. It's like using a secret code to unlock the chord's length every time! Plug in your numbers and watch the formula work its charm. Byju's: Chord Length Formula
  3. Perpendicular Bisector Theorem - If you drop a radius or diameter perpendicular to a chord, it slices the chord (and its corresponding arc) into two equal pieces. Picture it as a ninja star perfectly bisecting your chord and arc - sharp, precise, and evenly split! This theorem is your go‑to trick for showing equal parts. Online Math Learning: Perpendicular Bisector
  4. Congruent Chords and Arcs - In the land of circles, equal‑length chords cast equal‑length arcs, and vice versa. If two chords are twins, their arcs are twins too! This mirror‑like property makes it easy to spot equal angles and distances just by comparing chords and arcs. CliffsNotes: Arcs and Chords
  5. Equidistant Chords - Chords that sit the same distance from the circle's center are guaranteed to be the same length. Think of them as parallel friends hanging out at the exact same "altitude" inside the circle. This handy fact helps you spot equal chords without ever measuring them! Online Math Learning: Equidistant Chords
  6. Angles Formed by Intersecting Chords - When two chords cross inside a circle, the angle they make equals half the sum of the arcs they intercept. It's like mixing two smoothie flavors and then dividing the taste by two - math style! Use this rule to calculate those mysterious intersection angles. Math Warehouse: Intersecting Chords Angles
  7. Intersecting Chords Theorem - If two chords intersect, the product of the segments on one chord equals the product of the segments on the other: (segment 1 × segment 2) = (segment 3 × segment 4). It's classic balance - whatever you multiply on one side, you get on the other! Wikipedia: Intersecting Chords Theorem
  8. Parallel Chords and Arcs - Parallel chords in a circle catch congruent arcs between them. Picture two railway tracks on a circular loop - whatever "span" one track covers on the circle, the other covers the same! This is perfect for spotting equal measurements at a glance. MathBits Notebook: Parallel Chords
  9. Central Angles and Chords - Equal chords subtend equal central angles at the circle's heart, and equal central angles subtend equal chords. It's a two‑way street: chords and angles keep each other in check. Use this to switch between angles and lengths like a math ninja! Online Math Learning: Central Angles & Chords
  10. Angle Subtended by a Chord - The angle a chord makes at the circle's center is twice the angle it makes anywhere else on the circle's circumference. Think of it as your chord shouting, "I'm twice as loud at the center!" This rule is golden for solving tricky inscribed angle problems. Byju's: Angle Subtended by a Chord
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