MCQs / Mathematics / Construction of Triangles

Construction of Triangles MCQs — Mathematics

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.

1. How many triangles ABC can be constructed with angle A = 30^circ, b = AC = 10 cm and a = BC = 5 cm?

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

Answer: B) 1

Explanation: The height from C is h = bsin A = 10 12 = 5 cm. Since a = h, the arc just touches the base: one right-angled triangle. SLO M-09-B-30 · Knowledge

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2. Which instruments are needed to construct a triangle given two sides and an included angle of 70^circ?

  1. compass only
  2. ruler, compass and protractor
  3. set-square only
  4. ruler only

Answer: B) ruler, compass and protractor

Explanation: A protractor is needed for 70^circ (it cannot be made with a compass alone). SLO M-09-B-28 · Knowledge

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3. Is the data PQ = 4 cm, QR = 6 cm, angle R = 75^circ an example of SAS (two sides and the included angle)?

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

Answer: C) 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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4. Can a triangle be constructed with two angles 80^circ and 110^circ?

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

Answer: B) No

Explanation: The two angles add to 190^circ, which is not less than 180^circ, so a triangle cannot be made. SLO M-09-B-29 · Knowledge

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5. The three angle bisectors of a triangle meet at the:

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

Answer: B) incentre

Explanation: The incentre is equidistant from all three sides. SLO M-09-B-31 · Understanding

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6. Is the data AB = 5 cm, BC = 7 cm, angle B = 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. SAS needs three sides
  4. Only if the angle is 90^circ

Answer: A) Yes

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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7. A triangle has base 6 cm with base angles 45^circ and 45^circ. The triangle is:

  1. an isosceles right triangle
  2. equilateral
  3. isosceles with a 60^circ angle
  4. scalene and obtuse

Answer: A) an isosceles right triangle

Explanation: The third angle is 90^circ and the two base angles are equal. SLO M-09-B-29 · Knowledge

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8. Given two sides and the included angle, the number of different triangles that can be constructed is:

  1. exactly one
  2. none
  3. two
  4. infinitely many

Answer: A) exactly one

Explanation: SAS data fixes the triangle completely (SAS congruence). SLO M-09-B-28 · Knowledge

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9. A triangle is constructed with two sides of 5 cm and an included angle of 60^circ. The triangle is:

  1. scalene
  2. equilateral
  3. obtuse
  4. right-angled

Answer: B) equilateral

Explanation: Two equal sides with 60^circ between them force the other angles to be 60^circ each. SLO M-09-B-28 · Knowledge

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10. A triangle is to be constructed with angle A = 50^circ and angle B = 60^circ. The third angle will be:

  1. 250^circ
  2. 110^circ
  3. 80^circ
  4. 70^circ

Answer: D) 70^circ

Explanation: angle C = 180^circ - 50^circ - 60^circ = 70^circ. SLO M-09-B-29 · Knowledge

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11. Two sides 4 cm and 6 cm with an included angle of 180^circ would give:

  1. an equilateral triangle
  2. a right triangle
  3. no triangle (the points are in a straight line)
  4. two triangles

Answer: C) no triangle (the points are in a straight line)

Explanation: An angle of 180^circ is a straight angle; the vertices are collinear. SLO M-09-B-28 · Knowledge

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12. A triangle is to be constructed with angle A = 90^circ and angle B = 25^circ. The third angle will be:

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

Answer: C) 65^circ

Explanation: angle C = 180^circ - 90^circ - 25^circ = 65^circ. SLO M-09-B-29 · Knowledge

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13. How many triangles ABC can be constructed with angle A = 30^circ, b = AC = 12 cm and a = BC = 6 cm?

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

Answer: D) 1

Explanation: The height from C is h = bsin A = 12 12 = 6 cm. Since a = h, the arc just touches the base: one right-angled triangle. SLO M-09-B-30 · Knowledge

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14. The centroid of the triangle with vertices (1, 1), (4, 5) and (7, 0) is:

  1. (12, 6)
  2. ( 52, 3)
  3. (4, 2)
  4. (2, 4)

Answer: C) (4, 2)

Explanation: Centroid = ( x1 + x2 + x33, y1 + y2 + y33 ) = (4, 2). SLO M-09-B-31 · Understanding

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15. If the side given is not between the two given angles (AAS), we can still construct the triangle by first:

  1. finding the third angle, then using ASA
  2. bisecting the given side
  3. drawing a circle on the side
  4. using SSA

Answer: A) finding the third angle, then using ASA

Explanation: The angle sum gives the third angle, so the side now lies between two known angles. SLO M-09-B-29 · Knowledge

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16. Is the data PQ = 4 cm, PR = 6 cm, angle P = 75^circ an example of SAS (two sides and the included angle)?

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

Answer: D) Yes

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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17. The SSA case is called the ambiguous case because:

  1. the data may give two different triangles, one, or none
  2. it never gives a triangle
  3. it always gives two triangles
  4. the angle is always obtuse

Answer: A) the data may give two different triangles, one, or none

Explanation: Depending on the lengths, the arc can cut the base line twice, once or not at all. SLO M-09-B-30 · Knowledge

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18. Lines that pass through one common point are called:

  1. perpendicular
  2. concurrent
  3. parallel
  4. collinear

Answer: B) concurrent

Explanation: The point is the point of concurrency. SLO M-09-B-31 · Understanding

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19. A triangle is constructed with two sides 5 cm and 12 cm and an included angle of 90^circ. When measured, the third side should be:

  1. 14 cm
  2. 7 cm
  3. 13 cm
  4. 17 cm

Answer: C) 13 cm

Explanation: By Pythagoras the third side is 5^2 + 12^2 = 13 cm — a useful check on the accuracy of the construction. SLO M-09-B-28 · Knowledge

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20. The centroid of the triangle with vertices (2, 1), (8, 3) and (5, 8) is:

  1. (5, 2)
  2. (4, 5)
  3. (5, 4)
  4. (15, 12)

Answer: C) (5, 4)

Explanation: Centroid = ( x1 + x2 + x33, y1 + y2 + y33 ) = (5, 4). SLO M-09-B-31 · Understanding

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21. Two sides 7 cm and 7 cm with an included angle of 90^circ give a triangle whose other two angles are:

  1. 60^circ each
  2. 90^circ each
  3. 45^circ each
  4. 30^circ and 60^circ

Answer: C) 45^circ each

Explanation: It is an isosceles right triangle. SLO M-09-B-28 · Knowledge

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22. An altitude of a triangle is:

  1. the longest side
  2. a bisector of an angle
  3. a perpendicular from a vertex to the opposite side (or its extension)
  4. a line from a vertex to the midpoint of the opposite side

Answer: C) a perpendicular from a vertex to the opposite side (or its extension)

Explanation: Altitudes measure the heights of the triangle. SLO M-09-B-31 · Understanding

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23. In ASA data for triangle DEF the angles given are angle D and angle E. The side given must be:

  1. overlineEF
  2. overlineDE
  3. overlineDF
  4. any side

Answer: B) overlineDE

Explanation: In ASA the side lies between the two given angles, i.e. joins D and E. SLO M-09-B-29 · Knowledge

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24. How many triangles ABC can be constructed with angle A = 30^circ, b = AC = 8 cm and a = BC = 3 cm?

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

Answer: D) 0

Explanation: The height from C is h = bsin A = 8 12 = 4 cm. Since a = 3 < h, the arc cannot reach the base: no triangle. SLO M-09-B-30 · Knowledge

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25. In ASA data for triangle KLM the angles given are angle K and angle L. The side given must be:

  1. any side
  2. overlineKL
  3. overlineLM
  4. overlineKM

Answer: B) overlineKL

Explanation: In ASA the side lies between the two given angles, i.e. joins K and L. SLO M-09-B-29 · Knowledge

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