All questions
- Find the slope of the line passing through the points A(2, 3), B(4, 7). Answer: Slope = 2 m = y₂ - y₁/x₂ - x₁ = 7 - 3/4 - 2 = 4/2 = 2 . 1
- Find the slope of the line passing through the points P(-3, 1), Q(5, -2). Answer: Slope = -3/8 Slope =y₂-y₁/x₂-x₁ . Slope = -2-1/5-(-3)=-3/8 . 1
- Find the slope of the line passing through the points C(5, -2), D(7, 3). Answer: Slope = 5/2 Slope =y₂-y₁/x₂-x₁ . Slope = 3-(-2)/7-5=5/2 . 1
- Find the slope of the line passing through the points L(-2, -3), M(-6, -8). Answer: Slope = 5/4 Slope =y₂-y₁/x₂-x₁ . Slope = -8-(-3)/-6-(-2)=-5/-4=5/4 . 1
- Find the slope of the line passing through the points E(-4, -2), F(6, 3). Answer: Slope = 1/2 Slope =y₂-y₁/x₂-x₁ . Slope = 3-(-2)/6-(-4)=5/10=1/2 . 1
- Find the slope of the line passing through the points T(0, -3), S(0, 4). Answer: The slope cannot be determined. x₂-x₁=0-0=0 , so 4-(-3)/0 is not defined. Both points lie on the Y-axis, and a line parallel to (her... 1
- Determine whether the following points are collinear: A(-1, -1), B(0, 1), C(1, 3). Answer: The points are collinear. Slope =y₂-y₁/x₂-x₁ . Slope of AB = 1+1/0+1=2 Slope of BC = 3-1/1-0=2 The slopes are equal and the two segm... 1
- Determine whether the following points are collinear: D(-2, -3), E(1, 0), F(2, 1). Answer: The points are collinear. Slope =y₂-y₁/x₂-x₁ . Slope of DE = 0+3/1+2=1 Slope of EF = 1-0/2-1=1 The slopes are equal and the two segm... 1
- Determine whether the following points are collinear: L(2, 5), M(3, 3), N(5, 1). Answer: The points are not collinear. Slope =y₂-y₁/x₂-x₁ . Slope of LM = 3-5/3-2=-2 Slope of MN = 1-3/5-3=-1 The slopes are not equal, so th... 1
- Determine whether the following points are collinear: P(2, -5), Q(1, -3), R(-2, 3). Answer: The points are collinear. Slope =y₂-y₁/x₂-x₁ . Slope of PQ = -3+5/1-2=-2 Slope of QR = 3+3/-2-1=-2 The slopes are equal and the two... 1
- Determine whether the following points are collinear: R(1, -4), S(-2, 2), T(-3, 4). Answer: The points are collinear. Slope =y₂-y₁/x₂-x₁ . Slope of RS = 2+4/-2-1=-2 Slope of ST = 4-2/-3+2=-2 The slopes are equal and the two... 1
- Determine whether the following points are collinear: A(-4, 4), K(-2, 5/2), N(4, -2). Answer: The points are collinear. Slope =y₂-y₁/x₂-x₁ . Slope of AK = 5/2-4/-2+4=-3/2/2=-3/4 Slope of KN = -2-5/2/4+2=-9/2/6=-3/4 The slopes... 1
- In the figure 3.87 circles with centres X and Y touch each other at point Z. A secant passing through Z intersects the circles at points A and B respectively. Prove that, radius XA ∥ radius YB. Fill in the blanks and complete the proof. Answer: ∠ XAZ = ∠ YBZ (alternate angles), so radius XA ∥ radius YB. Construction: Draw segments XZ and YZ. By the theorem of touching circle... 1
- In figure 3.88, circles with centres X and Y touch internally at point Z. Seg BZ is a chord of bigger circle and it itersects smaller circle at point A. Prove that, seg AX ∥ seg BY. Answer: ∠ XAZ = ∠ XZA = ∠ YBZ , which are corresponding angles, so AX ∥ BY. Draw seg XZ and seg YZ. By the theorem of touching circles, Z, X... 1
- In figure 3.89, line l touches the circle with centre O at point P. Q is the mid point of radius OP. RS is a chord through Q such that chords RS ∥ line l. If RS = 12 find the radius of the circle. Answer: Radius = 4√(3) cm OP ⊥ l (tangent ⊥ radius) and RS ∥ l , so OQ ⊥ RS. The perpendicular from the centre bisects the chord, so RQ = 6.... 1
- In figure 3.90, seg AB is a diameter of a circle with centre C. Line PQ is a tangent, which touches the circle at point T. seg AP ⊥ line PQ and seg BQ ⊥ line PQ. Prove that, seg CP ≅ seg CQ. Answer: CT ⊥ PQ and T is the midpoint of PQ, so CT is the perpendicular bisector of PQ and CP = CQ. Draw seg CT. CT ⊥ line PQ (radius ⊥ tang... 1
- Draw circles with centres A, B and C each of radius 3 cm, such that each circle touches the other two circles. Answer: Draw an equilateral triangle ABC of side 6 cm and draw circles of radius 3 cm with centres A, B, C. Each pair of circles must touch... 1
- Prove that any three points on a circle cannot be collinear. Answer: If three points of a circle were collinear, the centre would lie on two distinct parallel perpendicular bisectors, which is impossib... 1
- In figure 3.91, line PR touches the circle at point Q. (1) Sum of ∠TAQ and ∠TSQ? (2) Angles congruent to ∠AQP? (3) Angles congruent to ∠QTS? (4) ∠TAS = 65°, find ∠TQS and arc TS. (5) If ∠AQP = 42° and ∠SQR = 58° find ∠ATS. Answer: (1) 180^° (2) ∠ ASQ , ∠ ATQ (3) ∠ SQR , ∠ SAQ (4) ∠ TQS = 65^° , arc TS = 130^° (5) ∠ ATS = 100^° (1) □TAQS is cyclic and ∠ TAQ , ∠... 1
- In figure 3.92, O is the centre of a circle, chord PQ ≅ chord RS. If ∠POR = 70° and (arc RS) = 80°, find - (1) m(arc PR) (2) m(arc QS) (3) m(arc QSR) Answer: (1) 70^° (2) 130^° (3) 210^° (1) m(arc PR) = central angle ∠ POR = 70^° . (2) Chord PQ ≅ chord RS, so m(arc PQ) = m(arc RS) = 80^° .... 1
- In figure 3.93, m(arc WY) = 44°, m(arc ZX) = 68°, then (1) Find the measure of ∠ZTX. (2) If WT = 4.8, TX = 8.0, YT = 6.4, find TZ. (3) If WX = 25, YT = 8, YZ = 26, find WT. Answer: (1) 56^° (2) TZ = 6 (3) WT = 16 or 9 (1) For chords intersecting inside a circle, ∠ ZTX = 1/2[m(arc ZX) + m(arc WY)] = 1/2(68^° + 44... 1
- In figure 3.94, (1) m(arc CE) = 54°, m(arc BD) = 23°, find measure of ∠CAE. (2) If AB = 4.2, BC = 5.4, AE = 12.0, find AD (3) If AB = 3.6, AC = 9.0, AD = 5.4, find AE Answer: (1) 15.5^° (2) AD = 3.36 (3) AE = 6 (1) Angle between two secants meeting outside the circle: ∠ CAE = 1/2[m(arc CE) - m(arc BD)] = 1... 1
- In figure 3.95, chord EF ∥ chord GH. Prove that, chord EG ≅ chord FH. Fill in the blanks and write the proof. Answer: Alternate angles ∠ EFG = ∠ FGH give m(arc EG) = m(arc FH), so chord EG ≅ chord FH. Draw seg GF. ∠ EFG = ∠ FGH …… alternate angles (E... 1
- In figure 3.96 P is the point of contact. (1) If m(arc PR) = 140°, ∠POR = 36°, find m(arc PQ) (2) If OP = 7.2, OQ = 3.2, find OR and QR (3) If OP = 7.2, OR = 16.2, find QR. Answer: (1) 68^° (2) OR = 16.2, QR = 13 (3) QR = 13 (1) Angle between a tangent and a secant drawn from an external point: ∠ POR = 1/2[m(arc... 1