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Construction of Triangles MCQs — Mathematics (page 3)

87 multiple-choice questions on Construction of Triangles from Mathematics, each with the correct answer and, where available, a teacher's explanation. Read through them to revise the chapter, then practise the same questions one by one to test yourself.

51. In an obtuse-angled triangle, the circumcentre lies:

  1. on the longest side
  2. at a vertex
  3. outside the triangle
  4. inside the triangle

Answer: C) outside the triangle

Explanation: The perpendicular bisectors meet beyond the longest side. SLO M-09-B-31 · Understanding

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52. The centre of the circle that touches all three sides of a triangle (the inscribed circle) is the:

  1. centroid
  2. orthocentre
  3. incentre
  4. circumcentre

Answer: C) incentre

Explanation: It is equidistant from the sides, so it lies on all three angle bisectors. SLO M-09-B-31 · Understanding

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53. Which set of data is SAS for triangle XYZ?

  1. XY, YZ, angle Y
  2. XY, YZ, XZ
  3. XY, angle X, angle Y
  4. XY, YZ, angle X

Answer: A) XY, YZ, angle Y

Explanation: angle Y is between sides XY and YZ. SLO M-09-B-28 · Knowledge

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54. In SSA with acute angle A, if a = bsin A, the triangle formed is:

  1. obtuse at A
  2. equilateral
  3. right-angled at B
  4. impossible

Answer: C) right-angled at B

Explanation: The arc just touches the base line at the foot of the perpendicular from C. SLO M-09-B-30 · Knowledge

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55. To construct triangle XYZ given moverlineXY, moverlineYZ and the included angle, the angle needed is:

  1. angle X
  2. angle Y
  3. angle Z
  4. any angle of the triangle

Answer: B) angle Y

Explanation: The included angle lies between the two given sides; XY and YZ meet at Y. SLO M-09-B-28 · Knowledge

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56. A triangle is constructed with two sides 8 cm and 15 cm and an included angle of 90^circ. When measured, the third side should be:

  1. 18 cm
  2. 23 cm
  3. 7 cm
  4. 17 cm

Answer: D) 17 cm

Explanation: By Pythagoras the third side is 8^2 + 15^2 = 17 cm — a useful check on the accuracy of the construction. SLO M-09-B-28 · Knowledge

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57. The last step of an SAS construction is to:

  1. bisect each side
  2. join the free ends of the two sides to form the third side
  3. measure the angles
  4. draw the circumcircle

Answer: B) join the free ends of the two sides to form the third side

Explanation: Joining the two end points completes the triangle. SLO M-09-B-28 · Knowledge

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58. Can a triangle be constructed with two angles 120^circ and 59^circ?

  1. Only if one angle is 90^circ
  2. Yes
  3. No
  4. Only if the side is longer than 10 cm

Answer: B) Yes

Explanation: The two angles add to 179^circ, which is less than 180^circ, so a triangle can be made. SLO M-09-B-29 · Knowledge

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59. In SSA, if the given angle is obtuse and the side opposite it is shorter than the other given side:

  1. an equilateral triangle is formed
  2. two triangles can be constructed
  3. no triangle can be constructed
  4. one triangle can be constructed

Answer: C) no triangle can be constructed

Explanation: The side opposite an obtuse angle must be the longest side. SLO M-09-B-30 · Knowledge

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60. A triangle is to be constructed with angle A = 42^circ and angle B = 73^circ. The third angle will be:

  1. 115^circ
  2. 65^circ
  3. 75^circ
  4. 245^circ

Answer: B) 65^circ

Explanation: angle C = 180^circ - 42^circ - 73^circ = 65^circ. SLO M-09-B-29 · Knowledge

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61. In an equilateral triangle, the incentre, circumcentre, centroid and orthocentre:

  1. form a square
  2. lie on one side
  3. are all different
  4. all coincide

Answer: D) all coincide

Explanation: Every median is also an altitude, an angle bisector and a perpendicular bisector. SLO M-09-B-31 · Understanding

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62. Which point always lies inside every triangle?

  1. the orthocentre
  2. the circumcentre
  3. the centroid (and the incentre)
  4. the midpoint of the longest side

Answer: C) the centroid (and the incentre)

Explanation: Medians and angle bisectors stay inside the triangle. SLO M-09-B-31 · Understanding

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63. How many triangles ABC can be constructed with angle A = 90^circ, b = AC = 7 cm and a = BC = 5 cm?

  1. 3
  2. 0
  3. 2
  4. 1

Answer: B) 0

Explanation: With angle A = 90^circ (not acute), the side opposite A must be the longest. Here a = 5 < b = 7, so there is no triangle. SLO M-09-B-30 · Knowledge

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64. How many triangles ABC can be constructed with angle A = 30^circ, b = AC = 8 cm and a = BC = 4 cm?

  1. 0
  2. 1
  3. 2
  4. 3

Answer: B) 1

Explanation: The height from C is h = bsin A = 8 12 = 4 cm. Since a = h, the arc just touches the base: one right-angled triangle. SLO M-09-B-30 · Knowledge

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65. To construct triangle LMN given moverlineLM, moverlineMN and the included angle, the angle needed is:

  1. angle M
  2. angle N
  3. angle L
  4. any angle of the triangle

Answer: A) angle M

Explanation: The included angle lies between the two given sides; LM and MN meet at M. SLO M-09-B-28 · Knowledge

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66. SSA data with angle A = 90^circ and the side opposite A longer than the other given side gives:

  1. no triangle
  2. two triangles
  3. exactly one triangle
  4. an equilateral triangle

Answer: C) exactly one triangle

Explanation: With a right angle the hypotenuse must be the longest side; then the triangle is unique (RHS). SLO M-09-B-30 · Knowledge

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67. The three altitudes of a triangle meet at the:

  1. circumcentre
  2. orthocentre
  3. incentre
  4. centroid

Answer: B) orthocentre

Explanation: An altitude is the perpendicular from a vertex to the opposite side. SLO M-09-B-31 · Understanding

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68. An angle of 45^circ can be constructed with compass and ruler by:

  1. bisecting a 60^circ angle
  2. trisecting a 180^circ angle
  3. bisecting a 90^circ angle
  4. only with a protractor

Answer: C) bisecting a 90^circ angle

Explanation: Construct a right angle, then bisect it. SLO M-09-B-28 · Knowledge

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69. In an SAS construction, after drawing one side and the angle at one end, the next step is to:

  1. bisect the angle
  2. mark the second side along the arm of the angle using a compass
  3. draw the third side first
  4. draw a perpendicular bisector

Answer: B) mark the second side along the arm of the angle using a compass

Explanation: Measure the second side along the arm, then join the end points. SLO M-09-B-28 · Knowledge

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70. A triangle has base QR = 6 cm, angle Q = 30^circ and angle R = 90^circ. Then angle P is:

  1. 30^circ
  2. 60^circ
  3. 90^circ
  4. 120^circ

Answer: B) 60^circ

Explanation: 180^circ - 30^circ - 90^circ = 60^circ. SLO M-09-B-29 · Knowledge

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71. Is the data AB = 5 cm, BC = 7 cm, angle A = 50^circ an example of SAS (two sides and the included angle)?

  1. Yes
  2. No — the angle is not between the two given sides
  3. Only if the angle is 90^circ
  4. SAS needs three sides

Answer: B) No — the angle is not between the two given sides

Explanation: For SAS the given angle must be at the vertex where the two given sides meet. SLO M-09-B-28 · Knowledge

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72. A right triangle has legs 6 cm and 8 cm. The radius of its circumcircle is:

  1. 10 cm
  2. 6 cm
  3. 7 cm
  4. 5 cm

Answer: D) 5 cm

Explanation: The hypotenuse = 10 cm is a diameter, so R = 5 cm. SLO M-09-B-31 · Understanding

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73. The three perpendicular bisectors of the sides of a triangle meet at the:

  1. circumcentre
  2. centroid
  3. orthocentre
  4. incentre

Answer: A) circumcentre

Explanation: The circumcentre is equidistant from the three vertices. SLO M-09-B-31 · Understanding

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74. The centre of the circle through all three vertices of a triangle is the:

  1. incentre
  2. circumcentre
  3. centroid
  4. orthocentre

Answer: B) circumcentre

Explanation: It is equidistant from the vertices, so it lies on the perpendicular bisectors. SLO M-09-B-31 · Understanding

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75. Can a triangle be constructed with two angles 95^circ and 85^circ?

  1. Yes
  2. Only if the side is longer than 10 cm
  3. No
  4. Only if one angle is 90^circ

Answer: C) No

Explanation: The two angles add to 180^circ, which is not less than 180^circ, so a triangle cannot be made. SLO M-09-B-29 · Knowledge

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