9th grade · Math · Unit 9 of 18
Circle Properties
Chords, tangents and angles
What your child practices
Using the rules of circle geometry: a perpendicular from the center bisects a chord, a central angle is twice an inscribed angle on the same arc, inscribed angles on the same arc are equal, and a tangent meets the radius at 90°.
Alabama Course of Study content standard: GDA.37 · central and inscribed angles, chords, radii and tangents of circles
How we explain it
Children can open this short lesson before they practice.
- A chord joins two points on a circle. A tangent touches it at one point.
- A central angle has its corner at the center.
- An inscribed angle is half the central angle on the same arc.
Example: A central angle is 80°. What is the inscribed angle on the same arc? Inscribed = half of central. 80 ÷ 2 = 40. Answer: 40°
Circle Properties in Alabama
In the Alabama Course of Study, this is unit 9 of 18 in 9th grade Math, after Right-Triangle Trig and before Line & Rotation Symmetry.
Gradelings describes a child's practice on this unit as Beginning, Approaching, Meeting or Exceeding (Four practice steps). That reflects practice, not a report-card mark: the teacher decides the mark.
Sample questions
A taste of this unit. In the app, questions change every time and get easier or harder to match your child.
Question 1
Two inscribed angles stand on the same arc. One measures 46°. How big is the other, in degrees?
answer box°
46 °
Hint: Inscribed angles on the same arc are equal.
Question 2
An inscribed angle on an arc measures 32°. What is the central angle on the same arc, in degrees?
answer box°
64 °
Hint: The central angle is twice the inscribed angle: 2 × 32 = 64.
Question 3
A tangent touches a circle at T. O is the center and P is a point on the tangent. Angle OPT is 63°. What is angle TOP, in degrees?
answer box°
27 °
Hint: A tangent is perpendicular to the radius at the point of contact, so angle OTP is 90°. The angles in triangle OTP add to 180°: 180 − 90 − 63 = 27.
Question 4
A circle has radius 15 cm. A chord is 24 cm long. How far is the chord from the center, in cm?
answer boxcm
9 cm
Hint: The perpendicular from the center meets the chord at its midpoint, 12 cm from each end. Use a right triangle with hypotenuse 15 and one leg 12: 15² − 12² = 81, and √81 = 9.
Question 5
A central angle on an arc measures 124°. What is an inscribed angle on the same arc, in degrees?
answer box°
62 °
Hint: An inscribed angle is half the central angle: 124 ÷ 2 = 62.
Question 6
A circle has radius 10 cm. A chord is 16 cm long. How far is the chord from the center, in cm?
answer boxcm
6 cm
Hint: The perpendicular from the center meets the chord at its midpoint, 8 cm from each end. Use a right triangle with hypotenuse 10 and one leg 8: 10² − 8² = 36, and √36 = 6.