Class 9 Mathematics MCQ Constructions

Question : Arrange the following steps of construction while constructing a pair of tangents to circle, which are inclined to each other at an angle of 60° to a circle of radius 3 cm.Steps of ConstructionStep I : Draw any diameter AOB of this circle 
Step II : Draw AM ⊥ AB and CN ⊥ OC. Let AM and CN intersect each other at P. Then PA and PC are the desired tangents to the given circle, inclined at an angle of 60°.
Step III : Draw a circle with O as centre and radius 3 cm.
Step IV : Construct ∠BOC = 60° such that radius OC meets the circle at C.
(a) III, I, IV, II
(b) III, II, IV, I
(c) II, I, IV, III
(d) IV, II, III, I

Question : Arrange the following steps of construction while constructing a pair of tangents to a circle of radius 3 cm from a point 10 cm away from the centre of the circle.Steps of ConstructionStep I : Bisect the line segment OP and let the point of bisection be M.
Step II : Taking M as centre and OM as radius, draw a circle. Let it intersect the given circle at the point Q and R.
Step III : Draw a circle of radius 3 cm.
Step IV : Join PQ and PR.
Step V : Take an external point P which is 10 cm away from its centre. Join OP.
(a) III, V, I, II, IV
(b) III, I, V, IV, II
(c) III, V, I, IV, II
(d) III, V, II, I, IV

Question :Let ABC be a right triangle in which AB = 3 cm, BC = 4 cm and ∠B = 90°. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Given below are the steps of constructions of a pair of tangents from A to this circle. Which of the following steps is INCORRECT? Steps of ConstructionStep I : Draw DABC and perpendicular BD from B on AC.
Step II : Draw a circle with BC as a diameter. This circle will pass through D.
Step III : Let O be the mid-point of BC. Join AO.
Step IV : Draw a circle with AO as diameter. This circle cuts the circle drawn in step II at B and P. AO, AP and AB are desired tangents drawn from A to the circle passing through B, C and D.
(a) Only Step I
(b) Only Step II
(c) Only Step III
(d) Only Step IV

Question : Given below are the steps of construction of a pair of tangents to a circle of radius 6 cm from a point on the concentric circle of radius 8 cm. Find which of the following steps is INCORRECT? Steps of Construction Step I : Take a point O on the plane paper and draw a circle of radius OA = 6 cm. Also, draw a concentric circle of radius OB = 8 cm.
Step II : Find the mid-point A of OB and draw a circle of radius BA = AO. Suppose this circle intersects the circle of radius 6 cm at P and Q.
Step III : Join BP and BQ to get the desired tangents.
(a) Step I
(b) Step II
(c) Step I and Step II
(d) Step II and Step III

Question : Given below are the steps of construction of a pair of tangents to a circle of radius 6 cm which are inclined to each other at an angle of 60°. Find which of the following step is wrong? Steps of Construction
I. With centre O and radius = 6 cm, draw a circle.

II. Taking a point A on the circle and draw ∠AOB = 120°.
III. Draw a perpendicular on OA at A. Draw another perpendicular on OB at B.
IV. Let the two perpendiculars meet at C. Thus CA and CB are the two required tangents to the given circle which are inclined to each other at 120°.
(a) Only Step I
(b) Only Step II
(c) Only Step III
(d) Only Step IV

Question :Given below are the steps of construction of two tangents to the circle (without using the centre of the circle ) of radius 4 cm from point P. Which of the following steps is INCORRECT?Steps of ConstructionStep I : Draw a circle of radius 4 cm and take a point P outside the circle and draw a secant PAB, intersecting the circle at A and B. 
Step II : Produce AP to C such that AP = CP. Draw a semicircle with CB as diameter.
Step III : Draw PD ⊥ CB, intersecting the semicircle at D. With P as centre and PC as radius draw arcs to intersect the given circle at T and T′.
Step IV : Join PT and PT′. Then, PT and PT′ are the required tangents.
(a) Only Step I
(b) Both Step I and Step II
(c) Only Step III
(d) Both Step II and Step IV

Question : Arrange the following steps of construction for constructing a Δ ABC in which AB = 4 cm, ∠B = 60° and altitude CL = 3 cm and then construct Δ ADE similar to Δ ABC such that each side of Δ ADE is 3/2 times that of the corresponding side of Δ ABC.
Steps of Construction
Step I : Join CA. Thus, D ABC is obtained.

Step II : Draw DE||BC, cutting AC produced at E.
Step III : Extend AB to D such that
Step III : AD = 3/2 AB = (2/3X4)CM = 6CM
Step IV : Draw a line segment AB = 4 cm.
Step V : Draw a line GH||AB at a distance of 3 cm, intersecting BP at C.
Step VI : Construct ∠ABP = 60°
(a) IV, VI, V, I, III, II
(b) IV, V, VI, I, III, II
(c) IV, V, I, III, II, VI
(d) V, IV, VI, III, I, II

Question : Given below are the steps of construction a triangle ABC with side BC = 6 cm, ∠B = 60°, ∠A = 105°and a triangle whose sides are (3/2) times the corresponding sides of Δ ABC. Which of the following steps of construction is INCORRECT?

Steps of ConstructionStep I : Draw BC = 6 cm.
Step II : At B construct ∠CBX = 60° and at C construct ∠BCY = 180° – (60° – 105°) = 15° Suppose BX and CY intersect at A. Δ ABC so obtained is the given triangle.
Step III : Construct an obtuse angle ∠CBZ at B on opposite side of vertex A of Δ ABC.
Step IV : Mark-off three (greater 3 of 2 in 3/2) points B₁, B₂, B₃, on BZ such that BB₁ = B₁B₂ = B₂B₃.
Step V : Join B₂ (the second point) to C and draw a line through B₃ parallel to B₂C, intersecting the extended line Segment BC at C′.
Step VI : Draw a line through C′ parallel to CA intersecting the extended line segment BA at A′. Triangle A′BC′ so obtained is the required triangle such that A’B/AB = BC’/BC = A’C’/AC = 3/2(A) Step III
(b) Step IV
(c) Step V
(d) Step II

Question :Which of the following steps of construction is INCORRECT while dividing a line segment of length 3.2 cm in the ratio of 3 : 5 internally. Steps of Construction
Step I : Draw AB = 3.2 cm
Step II : Construct an acute ∠BAX.
Step III : On AX make 3 + 5 + 1 i.e. 9 equal parts and mark them as A₁, A₂, A₃, A4, ……. A9
Step IV : Join B to A₈. From A₃ draw A₃C parallel to A8B. Point C divides AB internally in the ratio 3 : 5.
Thus, AC : CB = 3 : 5.
(a) Step II
(b) Step III
(c) Step IV
(d) None of these

Question : Arrange the following steps of construction while constructing a triangle of scale AB = 2.3 cm, BC = 5 cm and AC = 2.9 cm such that each of its sides is 2/3 rd of the corresponding side of the ΔABC. Steps of Construction

Step I : On BE, cut off 3 equal parts making B₁, B₂ and B₃.
Step II : Now, draw C′A′ parallel to CA. Then, DA′BC′ is the required D whose sides are 2/3 rd of the corresponding sides of the ΔABC.
Step III : From point B draw an arc of 2.3 cm and from point C draw an arc of 2.9 cm cutting each other at point A.
Step IV : Take BC = 5 cm.
Step V : Join B₃C and from B₂ draw B₂C′ parallel to B₃C, such that BC′ is 2/3 of BC.
Step VI : On B make an acute ∠CBE downwards.
Step VII : Join AB and AC. Then ABC is the required triangle.
(a) IV, III, VII, I, VI, V, II
(b) IV, V, I, VI, III, VII, II
(c) IV, III, VII, VI, I, V, II
(d) IV, VII, III, VI, V, I, II

Question : Arrange the steps of construction while constructing pair of tangents to a circle of radius 5 cm from a point 12 cm away from its centre.Steps of ConstructionStep I : Join OA and bisect it. Let P is the mid-point of OA.
Step II : Join AB and AC. AB and AC are the required tangents. Length of tangents = 11 cm.
Step III : With O as centre, draw a circle of radius 5 cm.
Step IV : Taking P as centre and PO as radius, draw a circle intersecting the given circle at the points B and C.
Step V : Take a point A at a distance of 12 cm from O.
(a) III, V, I, IV, II
(b) III, V, IV, I, II
(c) II, V, IV, III, I
(d) III, IV, II, I, III

Question : Arrange the following steps of construction while dividing a line segment of length 8 cm internally in the ratio 3 : 4.Steps of ConstructionStep I : Draw a ray BY parallel to AX by making ∠ABY equal to ∠BAX.
Step II : Join A₃B₄. Suppose it intersects AB at a point P. Then, P is the point dividing AB internally in the ratio 3 : 4.
Step III : Draw the line segment AB of length 8 cm.
Step IV : Mark of three point A₁, A₂, A₃ on AX and 4 points B₁, B₂, B₃, B₄ on BY such that AA1 = A₁A₂ = A₂A₃ = BB₁ = B₁B₂ = B₂B₃ = B₃B₄.
Step V : Draw any ray AX making an acute angle ∠BAX with AB.
(a) III, V, I, II, IV
(b) III, IV, I, V, II
(c) III, I, V, IV, II
(d) III, V, I, IV, II

Question : Which of the following steps is INCORRECT to construct a tangent to the circle of radius 5 cm at the point P on it without using the centre of the circle.Steps of ConstructionStep I : Draw a circle of radius 5 cm.
Step II : Mark a point P on it.
Step III : Draw any chord PQ.
Step IV : Take a point R in the minor arc QP.
Step V : Join PR and RQ.
Step VI : Make ∠QPT = ∠PRQ.
Step VII : Produce TP to T . Then, PT is the required tangent at P.
(a) Step II
(b) Step IV
(c) Step VI
(d) None of these