Circles: Chords, Arcs and Inscribed Angles: Geometry Math Test Answer Key – PDF Download

Geometry Circles Angles Test Answer Key: questions 1–5

Question 1: Similar circles (2 points)

1. \(\frac{15}{6}=\frac{5}{2}\). Scale factor \(\frac{5}{2}\)
2. Circumferences scale by the same factor: \(12\pi\cdot\frac{5}{2}=30\pi\). \(30\pi\)

Scoring: 1 point for each part.

Question 2: Central and inscribed angles (2 points)

1. A central angle equals its arc. \(124^{\circ}\)
2. An inscribed angle is half its arc: \(\frac{124}{2}=62\). \(62^{\circ}\)

Scoring: 1 point for each angle.

Question 3: Inscribed angle algebra (2 points)

\(5x+2=\frac{1}{2}(12x-12)\) gives \(5x+2=6x-6\), so \(x=8\). The angle is \(5(8)+2=42^{\circ}\) (the arc is \(84^{\circ}\)).

\(x=8\) and the angle measures \(42^{\circ}\).

Scoring: 1 point for a correct equation and \(x\); 1 point for the angle measure.

Question 4: Triangle in a semicircle (2 points)

1. An angle inscribed in a semicircle is a right angle. \(90^{\circ}\)
2. \(BC=\sqrt{26^{2}-10^{2}}=\sqrt{576}=24\). \(BC=24\)

Scoring: 1 point for naming the right angle with a reason; 1 point for the correct length.

Question 5: Tangent and radius (3 points)

1. A radius is perpendicular to a tangent at the point of tangency. \(90^{\circ}\)
2. \(PT=\sqrt{15^{2}-9^{2}}=\sqrt{144}=12\). \(PT=12\)
3. \(180^{\circ}-90^{\circ}-53^{\circ}=37^{\circ}\). \(37^{\circ}\)

Scoring: 1 point for each part.

Geometry Circles Angles Test Answer Key: questions 6–9

Question 6: Tangents from a point (2 points)

1. The angle at \(P\) and the minor arc sum to \(180^{\circ}\): \(180^{\circ}-64^{\circ}=116^{\circ}\). \(116^{\circ}\)
2. \(360^{\circ}-116^{\circ}=244^{\circ}\). \(244^{\circ}\)

Scoring: 1 point for each arc.

Question 7: Chord and center (2 points)

The perpendicular from the center bisects the chord, so half the chord is 20. \(d=\sqrt{25^{2}-20^{2}}=\sqrt{225}=15\).

The distance is 15 units.

Scoring: 1 point for using the bisected chord (half length 20); 1 point for the correct distance.

Question 8: Inscribed quadrilateral (3 points)

\(\angle A\) and \(\angle C\) are opposite: \((3x+15)+(5x-3)=180\), \(8x+12=180\), \(x=21\). \(\angle B\) and \(\angle D\) are opposite: \((4y+2)+(2y+22)=180\), \(6y+24=180\), \(y=26\).

\(\angle A=78^{\circ}\), \(\angle B=106^{\circ}\), \(\angle C=102^{\circ}\), \(\angle D=74^{\circ}\). Check: \(78+106+102+74=360\).

\(x=21\), \(y=26\); angles \(78^{\circ}\), \(106^{\circ}\), \(102^{\circ}\), \(74^{\circ}\).

Scoring: 1 point for \(x\); 1 point for \(y\); 1 point for all four angle measures.

Question 9: Skating rink (2 points)

1. Noah’s angle is inscribed in a semicircle. \(90^{\circ}\)
2. \(NB=\sqrt{60^{2}-36^{2}}=\sqrt{2304}=48\). 48 feet

Scoring: 1 point for the angle with its reason; 1 point for the correct distance.

Scoring summary

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