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13 multiple-choice questions and 15 flashcards on Circles, about 9% of the High School Geometry bank. Every one carries a written rationale.
Circles is one of 13 chapters in CoStudy's High School Geometry bank, and it holds 13 of the bank's 150 multiple-choice questions — roughly 9% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.
Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.
6 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.
Law of Sines: a/sin(A) = ?
Answer: C — b/sin(B) = c/sin(C) (ratios of sides to sine of opposite angles are equal)
Law of Sines: a/sin(A) = b/sin(C) = c/sin(B). Side over sine of opposite angle. Use when given AAS, ASA, or SSA. Law of Cosines for SAS or SSS.
An inscribed angle intercepts an arc of 140°. The inscribed angle measures:
Answer: B — 70°
A) Confused inscribed angle with CENTRAL angle (which equals the arc). B) Inscribed Angle Theorem: inscribed angle = (1/2) × intercepted arc = 140°/2 = 70°. C) Doubled the arc instead of halving. D) 180° − 140°, an unrelated computation.
Two chords of a circle intersect inside. One chord is split into segments of 4 and 6; the other has one segment of length 3. The other segment is:
Answer: C — 8
A) Divided by both numbers. B) Multiplied without dividing. C) Intersecting Chords (Power of a Point): 4·6 = 3·x ⇒ 24 = 3x ⇒ x = 8. D) Forgot to divide.
Two triangles are similar by AA (angle-angle). What can we conclude about corresponding sides?
Answer: A — Proportional (ratio of corresponding sides is constant — the scale factor)
Similar triangles: corresponding angles congruent, corresponding sides proportional. Scale factor = ratio of any pair of corresponding sides. AA, SAS, SSS are similarity criteria. Distinct from congruent (same size).
A circle has DIAMETER 14. Its area is:
Answer: D — 49π
A) Common trap — used the DIAMETER as the radius in πr². B) Used circumference of a unit circle by analogy — wrong. C) Used 2π·r — that's circumference, not area. D) radius = diameter/2 = 7, so area = π(7)² = 49π.
An inscribed angle subtending an arc of 100° measures:
Answer: D — 50°
A) Confused inscribed angle with central angle (which would equal the arc). D) Inscribed Angle Theorem: an inscribed angle is half the measure of its intercepted arc, so 100°/2 = 50°. C) Doubled the arc — the inverse mistake. B) 180° − 100°, an unrelated computation.
4 cards from the 15 in this chapter.
What is a 'secant' to a circle?
A line that intersects a circle at two points.
What's the relationship between a central angle and the arc it cuts off?
They have the same measure (in degrees).
What's the equation of a circle centered at (3, −2) with radius 5?
(x − 3)² + (y + 2)² = 25.
What's the relationship between a tangent and the radius at the point of tangency?
Perpendicular (90°).
These are a sample. The full Circles chapter runs 28 items with per-chapter progress tracking, on the web and in the iOS app.
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