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Form 6 Geometry and Trigonometry Synthesis Practice

Form 6 · 60 practice questions with instant marking and worked solutions.

Geometry and Trigonometry Synthesis is an important part of the Form 6 mathematics curriculum. This topic develops choosing trigonometric ratios, degree mode, bearings and 3-D contexts. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Geometry and Trigonometry Synthesis has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Geometry and Trigonometry Synthesis; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 8 of them are DSE Paper 1 style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Geometry and Trigonometry Synthesis set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Geometry and Trigonometry Synthesis should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    In a right triangle, the side opposite θ is 5 and the adjacent side is 12. Find $\tan\theta$. Choose the correct answer.
  2. 2 Basic
    In a right triangle, the side opposite θ is 6 and the adjacent side is 8. Find $\tan\theta$. Give the numerical value only.

  3. 3 Basic
    In a right triangle, the side opposite θ is 8 and the adjacent side is 15. Find $\tan\theta$. Show the essential working and give the numerical value.

    Reproducible diagram for Geometry and Trigonometry Synthesis; values and labels are stated in the question

  4. 4 Basic
    In Geometry and Trigonometry Synthesis, the value of $\tan\theta=\frac{3}{4}$ is 1. Is this statement true?
  5. 5 Basic
    $A$, $B$ and $C$ lie on a circle with centre $O$, $C$ on the major arc, and $\angle ACB=56^\circ$. Find $\angle AOB$ in degrees. Choose the correct answer.
  6. 6 Basic
    $ABCD$ is a cyclic quadrilateral with $\angle ABC=76^\circ$. Find $\angle ADC$ in degrees. Give the answer only.

  7. 7 Basic
    A circle has radius 5 cm and chord $AB$ is 3 cm from the centre $O$. Find the length of $AB$ in cm. Show the essential working and give the answer.

  8. 8 Basic
    $P$ is 29 cm from the centre $O$; the radius is 20 cm and $PT$ is a tangent from $P$ touching at $T$. The length of $PT$ in cm is 9. Is this true?
  9. 9 Basic
    $A$, $B$ and $C$ lie on a circle with centre $O$, $C$ on the major arc, and $\angle ACB=76^\circ$. Find $\angle AOB$ in degrees. Choose the correct answer.
  10. 10 Basic
    $ABCD$ is a cyclic quadrilateral with $\angle ABC=109^\circ$. Find $\angle ADC$ in degrees. Give the answer only.

  11. 11 Basic
    A circle has radius 25 cm and chord $AB$ is 7 cm from the centre $O$. Find the length of $AB$ in cm. Show the essential working and give the answer.

  12. 12 Basic
    $P$ is 17 cm from the centre $O$; the radius is 8 cm and $PT$ is a tangent from $P$ touching at $T$. The length of $PT$ in cm is 9. Is this true?
  13. 13 Standard
    In a right triangle, the side opposite θ is 10 and the adjacent side is 24. Find $\tan\theta$. Choose the correct answer.
  14. 14 Standard
    In a right triangle, the side opposite θ is 12 and the adjacent side is 16. Find $\tan\theta$. Give the numerical value only.

  15. 15 Standard
    In a right triangle, the side opposite θ is 16 and the adjacent side is 30. Find $\tan\theta$. Show the essential working and give the numerical value.

  16. 16 Standard
    In Geometry and Trigonometry Synthesis, the value of $\tan\theta=\frac{6}{8}$ is 3/4. Is this statement true?
  17. 17 Standard
    Consider the points $A(3,2)$ and $B(10,-22)$. Find the length of $AB$. Choose the correct answer.
  18. 18 Standard
    A cuboid measures 7 cm by 24 cm by 60 cm. Find the length of its space diagonal in cm. Give the answer only.

  19. 19 Standard
    Circle $C$ has equation $x^2+y^2+6x+4y-68=0$. Find the radius of $C$. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    $AB$ is a chord of a circle with centre $O$ and radius 15 cm, and $O$ is 9 cm from $AB$. $M$ is the midpoint of $AB$. Answer each part; give the answer only.
    1. (a) Find $AM$ in cm.
    2. (b) Find $AB$ in cm.
    3. (c) Find the area of $\triangle OAB$ in cm².
    4. (d) Find the greatest distance from a point on the major arc to $AB$ in cm.
  21. 21 Standard
    Consider the points $A(5,-4)$ and $B(25,-25)$. Find the length of $AB$. Choose the correct answer.
  22. 22 Standard
    Circle $C$ has equation $x^2+y^2-8x-33=0$. Find the radius of $C$. Give the answer only.

  23. 23 Standard
    Circle $C$ has equation $x^2+y^2+4y-5=0$. Find the radius of $C$. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    $AB$ is a chord of a circle with centre $O$ and radius 5 cm, and $O$ is 3 cm from $AB$. $M$ is the midpoint of $AB$. Answer each part; give the answer only.
    1. (a) Find $AM$ in cm.
    2. (b) Find $AB$ in cm.
    3. (c) Find the area of $\triangle OAB$ in cm².
    4. (d) Find the greatest distance from a point on the major arc to $AB$ in cm.
  25. 25 Advanced
    In a right triangle, the side opposite θ is 15 and the adjacent side is 36. Find $\tan\theta$. Choose the correct answer.
  26. 26 Advanced
    In a right triangle, the side opposite θ is 18 and the adjacent side is 24. Find $\tan\theta$. Give the numerical value only.

  27. 27 Advanced
    In a right triangle, the side opposite θ is 24 and the adjacent side is 45. Find $\tan\theta$. Show the essential working and give the numerical value.

  28. 28 Advanced
    In Geometry and Trigonometry Synthesis, the value of $\tan\theta=\frac{9}{12}$ is 1. Is this statement true?
  29. 29 Advanced
    In $\triangle ABC$, $CA=5$, $CB=6$ and $\angle C=60^\circ$. Find the value of $AB^2$. Choose the correct answer.
  30. 30 Advanced
    $A$, $B$ and $C$ lie on a circle with centre $O$, $C$ on the major arc, and $\angle ACB=40^\circ$. Find $\angle AOB$ in degrees. Give the answer only.

  31. 31 Advanced
    $ABCD$ is a cyclic quadrilateral with $\angle ABC=91^\circ$. $\angle ADC$ in degrees is 89. Is this true?
  32. 32 Advanced DSE long question
    Cuboid $ABCDEFGH$ has base $ABCD$ measuring 12 cm by 16 cm and height 21 cm. Answer each part; give the answer only.
    1. (a) Find the base diagonal $AC$ in cm.
    2. (b) Find the space diagonal $AG$ in cm.
    3. (c) Find the tangent of the angle between $AG$ and the base (as a fraction in lowest terms).
  33. 33 Advanced
    In $\triangle ABC$, $CA=7$, $CB=8$ and $\angle C=60^\circ$. Find the value of $AB^2$. Choose the correct answer.
  34. 34 Advanced
    $A$, $B$ and $C$ lie on a circle with centre $O$, $C$ on the major arc, and $\angle ACB=68^\circ$. Find $\angle AOB$ in degrees. Give the answer only.

  35. 35 Advanced
    $ABCD$ is a cyclic quadrilateral with $\angle ABC=103^\circ$. $\angle ADC$ in degrees is 77. Is this true?
  36. 36 Advanced DSE long question
    Cuboid $ABCDEFGH$ has base $ABCD$ measuring 8 cm by 15 cm and height 144 cm. Answer each part; give the answer only.
    1. (a) Find the base diagonal $AC$ in cm.
    2. (b) Find the space diagonal $AG$ in cm.
    3. (c) Find the tangent of the angle between $AG$ and the base (as a fraction in lowest terms).
  37. 37 Challenging
    In a right triangle, the side opposite θ is 20 and the adjacent side is 48. Find $\tan\theta$. Choose the correct answer.
  38. 38 Challenging
    In a right triangle, the side opposite θ is 24 and the adjacent side is 32. Find $\tan\theta$. Give the numerical value only.

  39. 39 Challenging
    In a right triangle, the side opposite θ is 32 and the adjacent side is 60. Find $\tan\theta$. Show the essential working and give the numerical value.

  40. 40 Challenging
    In Geometry and Trigonometry Synthesis, the value of $\tan\theta=\frac{12}{16}$ is 3/4. Is this statement true?
  41. 41 Challenging
    A circle has radius 29 cm and chord $AB$ is 20 cm from the centre $O$. Find the length of $AB$ in cm. Choose the correct answer.
  42. 42 Challenging
    Consider the points $A(5,-4)$ and $B(13,-10)$. Find the length of $AB$. Show the essential working and give the answer.

  43. 43 Challenging
    Consider the points $A(0,-2)$ and $B(3,-6)$. The length of $AB$ is 7. Is this true?
  44. 44 Challenging DSE long question
    In $\triangle ABC$, $CA=16$ cm, $CB=5$ cm and $\angle ACB=120^\circ$. Its area is written as $k\sqrt{3}$ cm². Answer each part; give the answer only.
    1. (a) Find $AB$ in cm.
    2. (b) Find $k$.
    3. (c) Find the perimeter of $\triangle ABC$ in cm.
  45. 45 Challenging
    A circle has radius 15 cm and chord $AB$ is 9 cm from the centre $O$. Find the length of $AB$ in cm. Choose the correct answer.
  46. 46 Challenging
    $P$ is 29 cm from the centre $O$; the radius is 20 cm and $PT$ is a tangent from $P$ touching at $T$. Find the length of $PT$ in cm. Show the essential working and give the answer.

  47. 47 Challenging
    Consider the points $A(5,1)$ and $B(-11,13)$. The length of $AB$ is 28. Is this true?
  48. 48 Challenging DSE long question
    In $\triangle ABC$, $CA=7$ cm, $CB=15$ cm and $\angle ACB=60^\circ$. Its area is written as $k\sqrt{3}$ cm². Answer each part; give the answer only.
    1. (a) Find $AB$ in cm.
    2. (b) Find $k$ (as a fraction in lowest terms).
    3. (c) Find the perimeter of $\triangle ABC$ in cm.
  49. 49 Elite
    In a right triangle, the side opposite θ is 25 and the adjacent side is 60. Find $\tan\theta$. Choose the correct answer.
  50. 50 Elite
    In a right triangle, the side opposite θ is 30 and the adjacent side is 40. Find $\tan\theta$. Give the numerical value only.

  51. 51 Elite
    In a right triangle, the side opposite θ is 40 and the adjacent side is 75. Find $\tan\theta$. Show the essential working and give the numerical value.

  52. 52 Elite
    In Geometry and Trigonometry Synthesis, the value of $\tan\theta=\frac{15}{20}$ is 1. Is this statement true?
  53. 53 Elite
    A cuboid measures 3 cm by 4 cm by 12 cm. Find the length of its space diagonal in cm. Give the answer only.

  54. 54 Elite
    Circle $C$ has equation $x^2+y^2+8x+2y+1=0$. Find the radius of $C$. Show the essential working and give the answer.

  55. 55 Elite
    In $\triangle ABC$, $CA=3$, $CB=8$ and $\angle C=60^\circ$. The value of $AB^2$ is 49. Is this true?
  56. 56 Elite DSE long question
    $ABCD$ is a cyclic quadrilateral inscribed in a circle with centre $O$. $\angle ABC=52^\circ$, $\angle BAD=64^\circ$, and $E$ lies on $AD$ produced. Answer each part; give the answer only.
    1. (a) Find $\angle ADC$ in degrees.
    2. (b) Find $\angle BCD$ in degrees.
    3. (c) Find $\angle CDE$ in degrees.
    4. (d) Find the non-reflex angle $\angle AOC$ in degrees.
  57. 57 Elite
    Circle $C$ has equation $x^2+y^2+4x+12y+24=0$. Find the radius of $C$. Give the answer only.

  58. 58 Elite
    Circle $C$ has equation $x^2+y^2+4x-10y+20=0$. Find the radius of $C$. Show the essential working and give the answer.

  59. 59 Elite
    In $\triangle ABC$, $CA=8$, $CB=5$ and $\angle C=60^\circ$. The value of $AB^2$ is 49. Is this true?
  60. 60 Elite DSE long question
    $ABCD$ is a cyclic quadrilateral inscribed in a circle with centre $O$. $\angle ABC=74^\circ$, $\angle BAD=76^\circ$, and $E$ lies on $AD$ produced. Answer each part; give the answer only.
    1. (a) Find $\angle ADC$ in degrees.
    2. (b) Find $\angle BCD$ in degrees.
    3. (c) Find $\angle CDE$ in degrees.
    4. (d) Find the non-reflex angle $\angle AOC$ in degrees.

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Geometry and Trigonometry Synthesis

This topic needs particular care with choosing trigonometric ratios, degree mode, bearings and 3-D contexts. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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