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GSE Geometry Unit 4 Circles and Arcs Review Guide Answer Key

Solve the GSE Geometry Unit 4 review with circle-angle theorems, intercepted arcs, cyclic quadrilaterals, arc length, and sector area.

Question

GSE Geometry Unit 4: Circles and Arcs Review GuideFor Problems 1–11, use the marked circle diagrams to find the requested value. The diagrams cover central angles, inscribed angles, tangent-chord angles, cyclic quadrilaterals, and angles formed inside or outside a circle.For Problems 12–18:12. Find the area of a quarter sector of radius 6.13. Find the length of a quarter-circle arc of radius 6.14. A radius-8 pizza is cut into 8 equal pieces. Find one piece's area.15. Find the outer-crust length of one piece from Problem 14.16. A circle has circumference 26π cm. Find its area.17. A sprinkler covers a 240° sector with radius 15 yd. Find the covered area.18. The radius from a clock's center to its numbers is 9 in. At 4:00, find the minor arc length between the hands and the area of the minor sector.

Problems 1–11: circles, arcs, and angles

  1. The central-angle relationships give x=22x=22, mBC^=88∘m\widehat{BC}=88^\circ, and mAC^=92∘m\widehat{AC}=92^\circ.
  2. Solving the equal-arc relationships gives x=35x=35, mBD^=135∘m\widehat{BD}=135^\circ, and mAC^=135∘m\widehat{AC}=135^\circ.
  3. An inscribed angle is half its intercepted arc, so m∠BAC=66∘m\angle BAC=66^\circ.
  4. The intercepted arc is twice the inscribed angle, so mBC^=54∘m\widehat{BC}=54^\circ.
  5. Half of the intercepted 86∘86^\circ arc gives m∠BAC=43∘m\angle BAC=43^\circ.
  6. The tangent-chord angle is supplementary to 104∘104^\circ, so its intercepted arc is 2(180−104)2(180-104): x=152∘x=152^\circ.
  7. Inscribed angles intercepting the same arc are congruent: 3x=2x+133x=2x+13, so x=13x=13.
  8. Opposite angles of a cyclic quadrilateral are supplementary: x+77=180x+77=180, so x=103∘x=103^\circ.
  9. Apply the exterior-angle half-difference rule to the marked arcs: x=35∘x=35^\circ.
  10. Applying the marked inscribed/exterior arc relationship gives x=81∘x=81^\circ.
  11. The exterior angle equals half the difference of the intercepted arcs:
x=315−452=135∘.x=\frac{315-45}{2}=135^\circ.

Problems 12–18: sector area and arc length

Use

L=θ360∘(2πr)andA=θ360∘(πr2).L=\frac{\theta}{360^\circ}(2\pi r) \qquad\text{and}\qquad A=\frac{\theta}{360^\circ}(\pi r^2).
  1. Quarter-sector area with r=6r=6: 9π≈28.279\pi\approx28.27 square units.
  2. Quarter-circle arc length with r=6r=6: 3π≈9.423\pi\approx9.42 units.
  3. One eighth of a pizza with r=8r=8: 8π≈25.13 in28\pi\approx25.13\text{ in}^2.
  4. Crust on one eighth of that pizza: 2π≈6.28 in2\pi\approx6.28\text{ in}.
  5. C=26πC=26\pi gives r=13r=13, so the area is 169π≈530.93 cm2169\pi\approx530.93\text{ cm}^2.
  6. A 240∘240^\circ sprinkler sector with r=15r=15: 150π≈471.24 yd2150\pi\approx471.24\text{ yd}^2.
  7. At 4:00, the hands form 120∘120^\circ. With r=9r=9, the arc length is 6π≈18.85 in6\pi\approx18.85\text{ in}, and the sector area is 27π≈84.82 in227\pi\approx84.82\text{ in}^2.

Rounding note

The exact answers in terms of π\pi are definitive. Some teacher-key decimals reflect substituting π≈3.14\pi\approx3.14: 28.26, 25.12, 530.66, 471, 18.84, and 84.78. For Problem 14, the key's 25.15 is a minor decimal discrepancy: 8π≈25.138\pi\approx25.13, while 8(3.14)=25.128(3.14)=25.12.

Evidence boundary

The problem order, diagrams, and teacher answers were checked against the teacher-hosted review guide and key. Exact pi values are calculated from the stated measurements. Most decimal differences in the key reflect use of pi approximately equal to 3.14; the key's Problem 14 value of 25.15 is a minor decimal discrepancy.

Sources

These references support the concepts and methods used in the explanation above.

GSE Geometry Unit 4 Circles and Arcs Answer Key | Verla