12 multiple-choice questions, progressively harder.
The given circle has radius 9.59.59.5 cm. A compass arc centered at A with opening OA selects B on the circle. What is the perimeter of triangle OAB?
Solution
Correct answer: C
Both OA and OB are radii of the given circle. AB is a radius of the compass circle with the same opening, so all three lengths are 9.5 cm.
Two consecutive radius-length compass steps locate B and then C after A. What is the smaller central angle AOC?
Correct answer: A
Each step gives an equilateral center triangle and a 60-degree turn. The first two turns total less than a straight angle, so they form the smaller angle.
After five consecutive radius-length steps from A around a circle, what additional forward turn returns the radius to its initial direction?
Correct answer: D
Five steps have used five equilateral center angles. The remaining part of a full turn is
This sixth step returns to the original ray and hence to its unique circle point A.
A given circle has diameter 20 cm. A compass step places a chord of length 10 cm on it. Which reason establishes a 60-degree smaller central angle?
Correct answer: B
The circle radius is half its diameter. The two radii and the constructed chord are all 10 cm, making an equilateral center triangle.
Its three equal angles share 180 degrees, giving a 60-degree center angle.
Consecutive hexagon vertices A and B have been marked with a fixed opening equal to the circle radius. An arc centered at B with that same opening meets the given circle at A and at a different point X. Which action continues the construction in the same direction?
A is the preceding vertex, so returning to it would undo the first step. The other crossing X belongs to the given circle and is one compass opening from B. Name it C and retain the opening for the next step.
An equilateral triangle is made by joining alternate vertices of a constructed regular hexagon in a circle of radius 3 cm. What is one triangle side?
A triangle side subtends 120 degrees. Halving its center triangle gives a 30-60-90 right triangle with hypotenuse 3 cm. Double the longer leg to recover the chord.
A constructed hexagon has vertices A, B, C, D, E, F in order. Which two vertices complete an equilateral triangle starting at B?
Starting at B, skip C to reach D, then skip E to reach F. Each pair of successive chosen vertices is two hexagon steps apart, including F back to B.
The three equal central angles give three equal chords.
For a square construction, diameter AC is 13 cm. Helper circles centered at A and C each use opening AC, meeting at U and V. What is AU + CU?
U lies on both helper circles. Each distance to a helper center equals the unchanged opening, 13 cm.
ABCD is the square obtained from two perpendicular diameters of circle O. What is angle ABO?
The center angle AOB is 90 degrees, and OA equals OB. The two base angles of triangle AOB are therefore equal.
A square is constructed inside a given circle of radius 9 cm. What is the length of one square diagonal?
Opposite square vertices are the endpoints of a constructed diameter. A diagonal is therefore twice the circle radius.
A square is constructed in a given circle of radius 424\sqrt242 cm. What is its side length?
Two neighboring radii are perpendicular and form the equal legs of a right triangle. The square side is their hypotenuse.
An equilateral triangle is constructed by alternate hexagon vertices in a circle of radius 232\sqrt323 cm. What is the triangle perimeter?
First find a triangle side from its 120-degree center triangle. Then count all three sides.
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