Reference / Curriculum / 3B / Chapter 12

3B · Chapter 12 · Solutions

Complete worked answer key

Answers and explanations for every original lesson practice set and chapter assessment.

Print this separately from student practice sheets. These are public worked solutions, not secure examination materials.

Circles

Prerequisite

  1. 3 — The triangle is a closed, three-sided polygon.
  2. 4 — A rectangle is a closed quadrilateral with four straight sides and four right angles. Trace the boundary once and count the straight segments.
  3. Yes — Its side and angle relationships remain the same.

Understanding checks

  1. 40 cm — A diameter joins two radii end to end through the center: 20 + 20 = 40 cm.
  2. 22 cm — The radius is half the diameter. 44 ÷ 2 = 22 cm.
  3. 50 cm — A diameter joins two radii end to end through the center: 25 + 25 = 50 cm.

Guided

  1. 14 cm — A diameter joins two radii end to end through the center: 7 + 7 = 14 cm.
  2. 26 cm — The radius is half the diameter. 52 ÷ 2 = 26 cm.
  3. 36 cm — A diameter joins two radii end to end through the center: 18 + 18 = 36 cm.
  4. 20 cm — The radius is half the diameter. 40 ÷ 2 = 20 cm.
  5. 13 cm — The radius is half the diameter. 26 ÷ 2 = 13 cm.

Independent

  1. 54 cm — A diameter joins two radii end to end through the center: 27 + 27 = 54 cm.
  2. 25 cm — The radius is half the diameter. 50 ÷ 2 = 25 cm.
  3. 18 cm — A diameter joins two radii end to end through the center: 9 + 9 = 18 cm.
  4. 5 cm — The radius is half the diameter. 10 ÷ 2 = 5 cm.
  5. 18 cm — The radius is half the diameter. 36 ÷ 2 = 18 cm.
  6. 32 cm — A diameter joins two radii end to end through the center: 16 + 16 = 32 cm.
  7. 10 cm — The radius is half the diameter. 20 ÷ 2 = 10 cm.
  8. 24 cm — A diameter joins two radii end to end through the center: 12 + 12 = 24 cm.

Challenge

  1. Second circle, by 1 cm — Their radii are 4 cm and 5 cm; changing diameter by 2 cm changes radius by 1 cm.
  2. 4 cm — The compass opening remains fixed while the pencil moves around the center.

Exit

  1. 3 cm — The radius is half the diameter. 6 ÷ 2 = 3 cm.
  2. 8 cm — A diameter joins two radii end to end through the center: 4 + 4 = 8 cm.
  3. 48 cm — A diameter joins two radii end to end through the center: 24 + 24 = 48 cm.

Angles

Prerequisite

  1. 3 cm — The radius is half the diameter. 6 ÷ 2 = 3 cm.
  2. 44 cm — A diameter joins two radii end to end through the center: 22 + 22 = 44 cm.
  3. 28 cm — The radius is half the diameter. 56 ÷ 2 = 28 cm.

Understanding checks

  1. acute — Compare 24° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. obtuse — Compare 137° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 19° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Guided

  1. obtuse — Compare 159° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. obtuse — Compare 145° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 49° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  4. obtuse — Compare 122° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  5. acute — Compare 48° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Independent

  1. acute — Compare 34° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. obtuse — Compare 164° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 40° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  4. obtuse — Compare 141° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  5. obtuse — Compare 156° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  6. acute — Compare 54° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  7. obtuse — Compare 109° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  8. obtuse — Compare 105° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Challenge

  1. No — Angle size depends on the turn between rays, not on how long they are drawn.
  2. Acute — 65° is below 90°, so the angle is acute; ray length is irrelevant.

Exit

  1. acute — Compare 62° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. obtuse — Compare 103° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. obtuse — Compare 101° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Right Angles

Prerequisite

  1. obtuse — Compare 142° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. acute — Compare 63° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 47° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Understanding checks

  1. acute — Compare 13° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. obtuse — Compare 123° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 57° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Guided

  1. obtuse — Compare 161° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. right — Compare 90° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. obtuse — Compare 162° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  4. acute — Compare 78° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  5. obtuse — Compare 145° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Independent

  1. obtuse — Compare 110° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. acute — Compare 33° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 45° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  4. obtuse — Compare 125° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  5. obtuse — Compare 95° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  6. obtuse — Compare 133° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  7. obtuse — Compare 137° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  8. obtuse — Compare 107° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Challenge

  1. No — Angle size depends on the turn between rays, not on how long they are drawn.
  2. Acute — 60° is below 90°, so the angle is acute; ray length is irrelevant.

Exit

  1. obtuse — Compare 163° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. acute — Compare 65° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 68° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Triangles

Prerequisite

  1. acute — Compare 65° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  2. obtuse — Compare 98° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. obtuse — Compare 110° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.

Understanding checks

  1. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  2. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  3. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Guided

  1. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  2. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  3. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  4. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  5. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Independent

  1. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  2. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  3. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  4. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  5. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  6. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  7. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  8. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Challenge

  1. Isosceles — It has exactly two equal sides. An equilateral triangle requires all three sides equal.
  2. No — The two shorter lengths total only 5, which cannot reach across a segment of length 8.

Exit

  1. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  2. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  3. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Properties of Triangles

Prerequisite

  1. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  2. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  3. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.

Understanding checks

  1. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  2. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  3. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.

Guided

  1. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  2. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  3. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  4. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  5. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.

Independent

  1. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  2. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  3. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  4. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  5. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  6. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  7. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  8. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Challenge

  1. Isosceles — It has exactly two equal sides. An equilateral triangle requires all three sides equal.
  2. No — The two shorter lengths total only 5, which cannot reach across a segment of length 8.

Exit

  1. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  2. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  3. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Properties of Quadrilaterals

Prerequisite

  1. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  2. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  3. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.

Understanding checks

  1. rectangle — The shape is a rectangle. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  2. rhombus — The shape is a rhombus. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  3. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.

Guided

  1. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  2. square — The shape is a square. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  3. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  4. rhombus — The shape is a rhombus. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  5. rhombus — The shape is a rhombus. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.

Independent

  1. square — The shape is a square. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  2. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  3. rhombus — The shape is a rhombus. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  4. rectangle — The shape is a rectangle. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  5. rhombus — The shape is a rhombus. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  6. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  7. rectangle — The shape is a rectangle. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  8. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.

Challenge

  1. No — A rotation preserves its equal side lengths and four right angles.
  2. Not a square; rhombus — A square requires right angles. A four-equal-sided quadrilateral without right angles is a rhombus.

Exit

  1. rectangle — The shape is a rectangle. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  2. square — The shape is a square. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  3. rhombus — The shape is a rhombus. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.

Using a Compass

Prerequisite

  1. rhombus — The shape is a rhombus. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  2. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  3. square — The shape is a square. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.

Understanding checks

  1. 10 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 10 mm from the center.
  2. 3 cm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 3 cm from the center.
  3. 17 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 17 mm from the center.

Guided

  1. 23 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 23 mm from the center.
  2. 2 cm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 2 cm from the center.
  3. 12 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 12 mm from the center.
  4. 22 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 22 mm from the center.
  5. 19 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 19 mm from the center.

Independent

  1. 4 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 4 mm from the center.
  2. 5 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 5 mm from the center.
  3. 3 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 3 mm from the center.
  4. 16 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 16 mm from the center.
  5. 2 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 2 mm from the center.
  6. 20 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 20 mm from the center.
  7. 25 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 25 mm from the center.
  8. 9 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 9 mm from the center.

Challenge

  1. Second circle, by 1 cm — Their radii are 5 cm and 6 cm; changing diameter by 2 cm changes radius by 1 cm.
  2. 5 cm — The compass opening remains fixed while the pencil moves around the center.

Exit

  1. 15 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 15 mm from the center.
  2. 14 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 14 mm from the center.
  3. 7 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 7 mm from the center.

Practice

  1. 14 cm — The radius is half the diameter. 28 ÷ 2 = 14 cm.
  2. acute — Compare 58° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 10° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  4. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  5. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  6. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  7. 3 cm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 3 cm from the center.
  8. 46 cm — A diameter joins two radii end to end through the center: 23 + 23 = 46 cm.
  9. acute — Compare 56° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  10. acute — Compare 60° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  11. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  12. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  13. rectangle — The shape is a rectangle. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  14. 3 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 3 mm from the center.
  15. 10 cm — The radius is half the diameter. 20 ÷ 2 = 10 cm.
  16. obtuse — Compare 154° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  17. Acute — 55° is below 90°, so the angle is acute; ray length is irrelevant.
  18. Second circle, by 1 cm — Their radii are 6 cm and 7 cm; changing diameter by 2 cm changes radius by 1 cm.

Chapter Review

  1. 58 cm — A diameter joins two radii end to end through the center: 29 + 29 = 58 cm.
  2. obtuse — Compare 170° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  3. acute — Compare 29° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  4. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  5. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  6. square — The shape is a square. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  7. 13 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 13 mm from the center.
  8. 30 cm — A diameter joins two radii end to end through the center: 15 + 15 = 30 cm.
  9. obtuse — Compare 119° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  10. obtuse — Compare 113° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  11. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  12. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  13. parallelogram — The shape is a parallelogram. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  14. 20 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 20 mm from the center.
  15. 15 cm — The radius is half the diameter. 30 ÷ 2 = 15 cm.
  16. acute — Compare 39° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  17. obtuse — Compare 136° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  18. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  19. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.
  20. rectangle — The shape is a rectangle. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types.
  21. 2 cm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 2 cm from the center.
  22. 40 cm — A diameter joins two radii end to end through the center: 20 + 20 = 40 cm.
  23. acute — Compare 59° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  24. acute — Compare 32° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse.
  25. Acute — 65° is below 90°, so the angle is acute; ray length is irrelevant.
  26. Second circle, by 1 cm — Their radii are 4 cm and 5 cm; changing diameter by 2 cm changes radius by 1 cm.

Practice Exam

  1. isosceles — The suggested answer is not correct. Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  2. obtuse — Compare 139° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  3. 22 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 22 mm from the center. (3 points: 1 point for the center, 1 for the accurate radius, 1 for a complete labeled circle.)
  4. 16 cm — The radius is half the diameter. 32 ÷ 2 = 16 cm. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  5. acute — The suggested answer is not correct. Compare 50° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  6. equilateral — Equilateral means three equal sides. These lengths satisfy the triangle inequality. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  7. scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  8. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  9. equilateral — The suggested answer is not correct. Equilateral means three equal sides. These lengths satisfy the triangle inequality. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  10. isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  11. acute — Compare 42° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  12. 9 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 9 mm from the center. (3 points: 1 point for the center, 1 for the accurate radius, 1 for a complete labeled circle.)
  13. obtuse — The suggested answer is not correct. Compare 101° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  14. acute — Compare 31° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  15. rectangle — The shape is a rectangle. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  16. 21 mm — The fixed compass opening is the radius. Keep its needle fixed and rotate the pencil to keep every perimeter point 21 mm from the center. (3 points: 1 point for the center, 1 for the accurate radius, 1 for a complete labeled circle.)
  17. square — The suggested answer is not correct. The shape is a square. Check right angles, equal-length sides and parallel lines; the description distinguishes it from the other listed types. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  18. 22 cm — The radius is half the diameter. 44 ÷ 2 = 22 cm. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  19. acute — Compare 78° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  20. obtuse — Compare 165° with 90°: angles smaller are acute, exactly 90° are right, and larger but less than 180° are obtuse. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  21. 9 cm — The suggested answer is not correct. The radius is half the diameter. 18 ÷ 2 = 9 cm. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  22. 48 cm — A diameter joins two radii end to end through the center: 24 + 24 = 48 cm. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  23. Acute — 60° is below 90°, so the angle is acute; ray length is irrelevant. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  24. Second circle, by 1 cm — Their radii are 5 cm and 6 cm; changing diameter by 2 cm changes radius by 1 cm. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)