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Circles and Construction Tools: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

Points $A$ and $B$ lie on a circle with center $O$ . Their distances from $O$ are $(3x+1)$ cm and $(5x-11)$ cm. Find the diameter's length.

Answer choices for question 1 A $19$ cm B $38$ cm C $6$ cm D $76$ cm
2

Diameter $AB$ contains center $O$ , with $AO=(2y+4)$ cm and $OB=(5y-8)$ cm. Find $AB$ .

Answer choices for question 2 A $12$ cm B $4$ cm C $48$ cm D $24$ cm
3

The equilateral-triangle procedure is attempted with $A=B$ . Which statement identifies the missing hypothesis and its consequence?

Answer choices for question 3 A The required positive length $AB$ is missing, so this procedure cannot form a triangle B Only a ruler is missing; all the circle conditions still hold C The construction gives a triangle with three zero-length sides D The compass radius becomes twice $AB$ and gives an ordinary triangle
4

An equilateral triangle made by the two-circle construction has perimeter $51$ cm. What compass radius was used?

Answer choices for question 4 A $25.5$ cm B $51$ cm C $17$ cm D $34$ cm
5

For an equilateral construction on $AB=9$ cm, which candidate has the required two circle distances?

Answer choices for question 5 A $AC=9$ cm and $BC=7$ cm B $AC=7$ cm and $BC=9$ cm C $AC=9$ cm and $BC=9$ cm D $AC=4.5$ cm and $BC=4.5$ cm
6

Two disks in the same plane share a center and have radii $5$ cm and $8$ cm. Where is every point on the smaller disk's boundary circle?

Answer choices for question 6 A Inside the larger disk and on the smaller disk boundary B On the larger circle and outside the smaller disk C Outside both disks D At the common center
7

An angle copy uses source circle radius $8$ cm and source chord length $6$ cm. The target circle at $P$ has already been drawn with radius $8$ cm and meets the baseline at $D$ . Which circle locates the next mark?

Answer choices for question 7 A Center $P$ , radius $6$ cm B Center $D$ , radius $8$ cm C Center $P$ , radius $16$ cm D Center $D$ , radius $6$ cm
8

Two source rays bound a reflex opening of $260^\circ$ . Their smaller opening is copied using the angle-copy procedure. What reflex opening do the target rays then bound?

Answer choices for question 8 A $100^\circ$ B $260^\circ$ C $130^\circ$ D $230^\circ$
9

A circle has center $O$ and positive radius $r$ , and $AB$ is a chord with distinct endpoints. Which additional fact permits the equation $AB=AO+OB=2r$ ?

Answer choices for question 9 A $O$ lies on segment $AB$ . B $AO=BO$ . C $O$ lies in the circle's plane. D AB is drawn horizontally.
10

A circle centered at $P$ is drawn with positive compass opening $UV$ , meeting ray $PX$ at $Q$ . Without changing the opening, a second circle is drawn centered at $Q$ , and $R$ is chosen on it. Which reason establishes $QR=UV$ ?

Answer choices for question 10 A The circles have the same center, so the two distances agree B Every chord is a radius, so the two distances agree C $PQ=UV$ and $QR=PQ$ , since both are radii drawn with the retained opening D Every point of the first circle lies on the second, so the two distances agree
11

An equilateral triangle is constructed on $AB=6$ cm. Point $M$ is the midpoint of $AB$ . What is $AM+BC$ ?

Answer choices for question 11 A $6$ cm B $9$ cm C $12$ cm D $15$ cm
12

Which statement matches the justifications established in this lesson?

Answer choices for question 12 A A protractor proves every ideal angle copy exactly B Equal-looking sides establish an equilateral triangle C Triangle angle sums are needed to justify the segment copy D The segment copy and equilateral construction follow from circle radii; the full angle-copy proof comes later

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