Loci Questions and Answers - Form 4 Topical Mathematics

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 Questions

  1.  
    1. Using a ruler, a pair of compasses only construct triangle XYZ such that XY = 6cm, YZ = 8cm and ∠XYZ = 75o
    2. Measure line XZ and ∠XZY
    3. Draw a circle that passes through X, Y and Z
    4. A point M moves such that it is always equidistant from Y and Z. construct the locus of M and define the locus
  2.  
    1. Construct a triangle ABC in which AB=6cm, BC = 7cm and angle ABC = 75o
      Measure
      1. Length of AC
      2. Angle ACB
    2. Locus of P is such that BP = PC. Construct P
    3. Construct the locus of Q such that Q is on one side of BC, opposite A and angle BQC = 30o
      1. Locus of P and locus of Q meet at X. Mark x
      2. Construct locus R in which angle BRC 120o
      3. Show the locus S inside triangle ABC such that XS ≥SR
  3. Use a ruler and compasses only for all constructions in this question.
      1. Construct a triangle ABC in which AB=8cm, and BC=7.5cm and ∠ABC=112½°
      2. Measure the length of AC
    1. By shading the unwanted regions show the locus of P within the triangle ABC such that
      1. AP ≤ BP
      2. AP >3cm
        Mark the required region as P
    2. Construct a normal from C to meet AB produced at D
    3. Locate the locus of R in the same diagram such that the area of triangle ARB is ¾ the area of the triangle ABC.
  4. On a line AB which is 10 cm long and on the same side of the line, use a ruler and a pair of compasses only to construct the following.
    1. Triangle ABC whose area is 20 cm2 and angle ACB = 90o
      1. The locus of a point P such that angle APB = 45o.
      2. Locate the position of P such that triangle APB has a maximum area and calculate this area.
  5. A garden in the shape of a polygon with vertices A, B, C, D and E. AB = 2.5m, AE = 10m, ED = 5.2M and DC=6.9m. The bearing of B from A is 030º and A is due to east of while D is due north of E, angle EDC = 110º,
    1. Using a scale of 1cm to represent 1m construct an accurate plan of the garden
    2. A foundation is to be placed near to CD than CB and no more than 6m from A,
      1. Construct the locus of points equidistant from CB and CD.
      2. Construct the locus of points 6m from A
      1. shade and label R ,the region within which the foundation could be placed in the garden
      2. Construct the locus of points in the garden 3.4m from AE.
      3.  Is it possible for the foundation to be 3.4m from AE and in the region?
  6.  
    1. Using a ruler and compasses only construct triangle PQR in which QR= 5cm, PR = 7cm and angle PRQ = 135°
    2. Determine ∠ PQR
    3. At P drop a perpendicular to meet QR produced at T
    4. Measure PT
    5. Locate a point A on TP produced such that the area of triangle AQR is equal to one and – a - half times the area of triangle PQR
    6. Complete triangle AQR and measure angle AQR
  7. Use ruler and a pair of compasses only in this question.
    1. Construct triangle ABC in which AB = 7 cm, BC = 8 cm and ABC = 60o.
    2. Measure
      1. side AC 
      2. ACB
    3. Construct a circle passing through the three points A, B and C. Measure the radius
    4. Construct ∆ PBC such that P is on the same side of BC as point A and PCB = ½ ACB, BPC = BAC measure PBC.
  8. Without using a set square or a protractor:-
    1. Construct triangle ABC in which BC is 6.7cm, angle ABC is 60o and ∠BAC is 90o.
    2. Mark point D on line BA produced such that line AD =3.5cm
    3. Construct:-
      1. A circle that touches lines AC and AD
      2. A tangent to this circle parallel to line AD
  9. Use a pair of compasses and ruler only in this question;
    1. Draw acute angled triangle ABC in which angle CAB = 37½o, AB = 8cm and CB = 5.4cm. Measure the length of side AC (hint 37½o = ½ x 75o)
    2. On the triangle ABC above:
      1. On the same side of AC as B, draw the locus of a point X so that angle AXC = 52½o
      2. Also draw the locus of another point Y, which is 6.8cm away from AC and on the same side as X
    3. Show by shading the region P outside the triangle such that angle APC ≥52 ½o and P is not less than 6.8cm away from AC

Answers

  1.  

    1.  loci ans1a
    2. XYZ = 42o + 1o
      XZ = 8.8 + 0.1 cm
    3. Bisecting any two sides
      Drawing the circle
    4. Perpendicular bisector of YZ
      Identification of locus of M
  2. AC = 8 cm ±0.1
    ∠ACB = 460 ±10
    1. AC = 12.9 ±0.1cm
      1. Line and well shaded B2
    2. h = 7 ±0.1
    3. ΔABC _____ Area = ½ x 8 x 7cm
      = 28cm
      i.e. ¾ x 28 = Area for ARB
      = 21cm
      i.e. ½ x 8 x h = 21
      h = 5.25
  3.  
    Loci ans4
    MP = 12 cm
    Area
    APB = ½ x 10 x 12 = 60 cm2
  4.  
    1.  
      loci ans 5i
    2. Yes
  5.  
    1.  
      loci ans6
    2. ∠ PQR = 26° + 1°
    3. 4.9 ± 0.1cm
    4. AT = u = 8.7cm
    5. ∠AQR = 37 ± 1
  6.  
    1. AB = 7 cm, BC = 8 cm
      ABC = 60o
    2. AC = 7.6 + 0.1 cm
      ABC = 53o ±0.1
    3. Perpendicular bisectors of any two sides.
      Circle drawn
      Radius = 4.4.±0.1. cm
    4. ACB bisected
      Bisection line drawn to cut circle at P
      BPC = BAC = 67o
      PBC = 88 ± 0.1o

      loci ans7
  7. B1 – Line AC
    B1 Line AB
    B1 AD
    B3 – Drawing correct circle
    B2- Tangent correctly drawn
    loci ans8
  8.  
    1. B1 for constructing 150o
      B1 for constructing 75oB1 for completing triangle ABC
      B1 for AC = 8.8 ± 0.1
      1. B1 For locating locus centre
        B1 for locus of X
      2. B1 for constructing arcs 6.8cm from AC
        B1 for locus Y
    2. B2 for shading the locus of P
      loci ans9

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