All questions
- Construct ΔPYQ such that, PY = 6.3 cm, YQ = 7.2 cm, PQ = 5.8 cm. If YZ/YQ = 6/5, then construct ΔXYZ similar to ΔPYQ. Answer: ΔXYZ has XY = 7.56 cm, YZ = 8.64 cm, XZ = 6.96 cm. Analysis: ΔXYZ ~ ΔPYQ (X↔P, Y↔Y, Z↔Q), so XY/PY=YZ/YQ=XZ/PQ=6/5 . ΔXYZ is bigger... 1
- Find the distance between the points A(2, 3), B(4, 1). Answer: AB = 2sqrt2 Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . AB=√((4-2)²+(1-3)²) =√(4+4)=sqrt8=2sqrt2 1
- Find the distance between the points P(-5, 7), Q(-1, 3). Answer: PQ = 4sqrt2 Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . PQ=√((-1+5)²+(3-7)²) =√(16+16)=√(32)=4sqrt2 1
- Find the distance between the points R(0, -3), S(0, 5/2). Answer: RS = 11/2 Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . Both points are on the Y-axis. RS=√(0²+(5/2+3)²) =√((11/2)²)=11/2 1
- Find the distance between the points L(5, -8), M(-7, -3). Answer: LM = 13 Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . LM=√((-7-5)²+(-3+8)²) =√(144+25)=√(169)=13 1
- Find the distance between the points T(-3, 6), R(9, -10). Answer: TR = 20 Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . TR=√((9+3)²+(-10-6)²) =√(144+256)=√(400)=20 1
- Sum of the squares of adjacent sides of a parallelogram is 130 sq.cm and length of one of its diagonals is 14 cm. Find the length of the other diagonal. Answer: The other diagonal is 8 cm. Sum of the squares of the diagonals of a parallelogram = sum of the squares of its four sides = 2 × (sum... 1
- In ΔABC, seg AD ⊥ seg BC, DB = 3CD. Prove that: 2AB² = 2AC² + BC² Answer: With BC = 4CD, Pythagoras theorem gives AB² - AC² = 8CD² = BC²/2 , hence 2AB² = 2AC² + BC² . BC = CD + DB = CD + 3CD = 4CD, so BC² =... 1
- In an isosceles triangle, length of the congruent sides is 13 cm and its base is 10 cm. Find the distance between the vertex opposite the base and the centroid. Answer: The distance is 8 cm. Let triangle ABC have AB = AC = 13 cm and BC = 10 cm, and let AD be the median to the base. In an isosceles tr... 1
- In a trapezium ABCD, seg AB ∥ seg DC, seg BD ⊥ seg AD, seg AC ⊥ seg BC, If AD = 15, BC = 15 and AB = 25. Find A(□ABCD) Answer: A(□ABCD) = 192 sq. units In right-angled triangle ADB ( ∠ ADB = 90^° ): BD² = 25² - 15² = 400, BD = 20 Draw DE ⊥ AB. The altitude on... 1
- In the figure 2.35, ΔPQR is an equilateral triangle. Point S is on seg QR such that QS = (1/3) QR. Prove that: 9 PS² = 7 PQ² Answer: With side a, PT² = 3a²/4 and ST = a/6 , so PS² = 7a²/9 , i.e. 9PS² = 7PQ² . Let PQ = QR = PR = a. In an equilateral triangle the per... 1
- Seg PM is a median of ΔPQR. If PQ = 40, PR = 42 and PM = 29, find QR. Answer: QR = 58 By Apollonius theorem, PQ² + PR² = 2PM² + 2QM² 1600 + 1764 = 2 × 841 + 2QM² 3364 - 1682 = 2QM² QM² = 841, QM = 29 M is the m... 1
- Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the distance between their centres. Answer: The distance between the centres is 1.3 cm. When two circles touch each other internally, the distance between their centres equals... 1
- Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance between their centres. Answer: The distance between the centres is 9.7 cm. When two circles touch each other externally, the distance between their centres equals... 1
- If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles touching each other - (i) externally (ii) internally. Answer: (i) Keep the centres 6.8 cm apart; (ii) keep the centres 1.2 cm apart. (i) Touching externally (distance between centres = 4 + 2.8 =... 1
- In fig 3.27, the circles with centres P and Q touch each other at R. A line passing through R meets the circles at A and B respectively. Prove that - (1) seg AP ∥ seg BQ, (2) ΔAPR ~ ΔRQB, and (3) Find ∠RQB if ∠PAR = 35° Answer: (1) and (2) are proved below; (3) ∠ RQB = 110^° . By the theorem of touching circles, P, R, Q are collinear. (1) ∠ PRA cong ∠ QRB (v... 1
- In fig 3.28 the circles with centres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm. Answer: CD = 4√(6) cm AB = 4 + 6 = 10 cm (circles touching externally). AC ⊥ l and BD ⊥ l (tangent ⊥ radius), so AC ∥ BD and □ACDB is a trap... 1
- In figure 3.37, points G, D, E, F are concyclic points of a circle with centre C. ∠ECF = 70°, m(arc DGF) = 200° find m(arc DE) and m(arc DEF). Answer: m(arc DE) = 90^° , m(arc DEF) = 160^° Arcs DGF and DEF together make the whole circle: m(arc DEF) = 360^° - 200^° = 160^° The measur... 1
- In fig 3.38 ΔQRS is an equilateral triangle. Prove that, (1) arc RS ≅ arc QS ≅ arc QR (2) m(arc QRS) = 240°. Answer: Congruent chords give congruent arcs, so each minor arc is 120^° and arc QRS = 240^° . (1) triangle QRS is equilateral, so chord RS... 1
- In fig 3.39 chord AB ≅ chord CD, Prove that, arc AC ≅ arc BD Answer: Congruent chords give arc ACB ≅ arc CBD; subtracting the common arc CB gives arc AC ≅ arc BD. Chord AB ≅ chord CD (given). therefore... 1
- In figure 3.56, in a circle with centre O, length of chord AB is equal to the radius of the circle. Find measure of each of the following. (1) ∠AOB (2) ∠ACB (3) arc AB (4) arc ACB. Answer: (1) 60^° (2) 30^° (3) 60^° (4) 300^° (1) OA = OB = AB (radii and given), so triangle AOB is equilateral and ∠ AOB = 60^° . (2) By th... 1
- In figure 3.57, □PQRS is cyclic. side PQ ≅ side RQ. ∠PSR = 110°, Find- (1) measure of ∠PQR (2) m(arc PQR) (3) m(arc QR) (4) measure of ∠PRQ Answer: (1) 70^° (2) 220^° (3) 110^° (4) 55^° (1) Opposite angles of a cyclic quadrilateral are supplementary: ∠ PQR = 180^° - 110^° = 70^°... 1
- □MRPN is cyclic, ∠R = (5x − 13)°, ∠N = (4x + 4)°. Find measures of ∠R and ∠N. Answer: ∠ R = 92^° , ∠ N = 88^° In □MRPN, ∠ R and ∠ N are opposite angles. Opposite angles of a cyclic quadrilateral are supplementary: (5x... 1
- In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that ∠RTS is an acute angle. Answer: One of seg TR, seg TS cuts the circle at a point Q; ∠ RQS = 90^° is an exterior angle of a triangle with ∠ RTS as a remote interior... 1