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Construction gives an insight on how various geometrical shapes are drawn with the help of different methods. Various geometrical tools such as: divider, compass, etc shall be used to construct various geometries such as: similar triangles, tangent to a circle, dividing a line segment, etc.
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Steps of Construction
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To divide the line segment internally in a ratio of m : n. Given is that both the numerals that are p and q are positive.
Steps of Construction
- Step 1. Firstly, draw a line segment of a given length, namely AB.
- Step 2. Now, make an acute angle with the line segment AB, namely via ray AX.
- Step 3. Along the ray AX, mark the points (m + n) as A1, A2 ….. Ap + q such that AA1 = A1A2…..= Am + n.
- Step 4. Next, join the coordinates of B Am + n.
- Step 5. Finally, through the point Am, draw a line parallel to Am + n. This should be made along by making an angle that is ultimately equal to that of an angle AAm + n B at point c.
- Hence, the point c is the point that divides a line segment internally.2)
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Construction of a tangent to a circle from a point that is given outside a circle.
Steps of Construction
- Step 1. Firstly, take any O as a center of a circle and let P be the exterior point from which the tangent is to be drawn.
- Step 2. Now, join the points PO and then bisect it. Then, take M as mid – point of PO.
- Step 3. Simultaneously, taking M as the centre of the circle and MO (or MP) as the radius, draw a circle.
- Step 4. So, now let the new circle intersect the given circle at the points Q and R.
- Step 5. Now, Join PQ and PR.
- Step 6. Finally, the PQ and PR are the required tangents to a circle.
Construction of a tangent to a circle from a point that is given on a circle.
Steps of Construction
- Step 1. Firstly, take any O as a center of a circle and let P be the exterior point from which the tangent is to be drawn.
- Step 2. Now, join the points PO and then cut the arcs of the required length and mark the arc outside the circle as R.
- Step 3. From the points O and R (From the arcs) bisect the line OR and name it as S.
- Step 4. Hence, the line extending from point S is the required tangent.
Construct a triangle similar to that of a given triangle, when m is less than that of n as given below.
Steps of Construction
- Step 1. Begin by constructing any triangle ABC.
- Step 2. Let AB be the base of a triangle. Subsequently, it could be even BC or AC.
- Step 3. Below the base AB, construct an acute angle as an angle BAX.
- Step 4. Now, mark n points as A1, A2, and so on at AX. This marking should be such that AA1 = A1A2 = A n – 1 An.
- Step 5. Now, for ease, join An B.
- Step 6. Finally, draw An B’ that is parallel to An B. This will meet AB at point B’.
- Step 7. Now, draw B’ C’ that will be parallel to CB, meeting AC at C’.
Hence, the triangle is ready.
Construct a triangle similar to that of a given triangle, when m is greater than that of n as given below.
Steps of Construction
- Step 1. Begin by constructing any triangle ABC.
- Step 2. Let AB be the base of a triangle. Subsequently, it could be even BC or AC.
- Step 3. Below the base AB, construct an acute angle as an angle BAX.
- Step 4. Now, mark m points as A1, A2, and so on at AX. This marking should be such that AA1 = A1A2 = A m – 1 Am.
- Step 5. Now, for ease, join An B.
- Step 6. Finally, draw a line from Am that is parallel to An B. This will meet AB at point B’.
- Step 7. Now, draw B’ C’ that will be parallel to CB, meeting AC at C’.
Hence, the triangle AB’C’ is ready.
Construction of a triangle similar to that of a given triangle, when m is greater than that of n as given below.
Steps of Construction
- Step 1. Begin by constructing any triangle ABC.
- Step 2. Let AB be the base of a triangle. Subsequently, it could be even BC or AC.
- Step 3. Below the base AB, construct an acute angle as an angle BAX.
- Step 4. Now, mark m points as A1, A2, and so on at AX. This marking should be such that AA1 = A1A2 = A m – 1 Am.
- Step 5. Now, for ease, join An B.
- Step 6. Finally, draw a line from Am that is parallel to An B. This will meet AB at point B’.
- Step 7. Now, draw B’ C’ that will be parallel to CB, meeting AC at C’.
Hence, the triangle AB’C’ is ready.
Sample Questions
Ques. Draw a line segment which shall have the length of 8 cm. Then, find a point M on it which divides it in the ratio of 3:5. (2 marks)
Ans:
Steps of construction:
1. Initially, draw a line segment, namely, AB of the given 8 cm.
2. Then, draw a ray, namely, AX, while making an acute angle downward with AB.
3. Mark the points as A1, A2, A3 … A8 on AX.
4. Marking of these points shall be in such a way that AA1 = A1A2 = A2A3 = ….., A7A8.
5. Then, join BA8.
6. Finally, at the end, draw a line parallel to BA8 through the point A3, to meet AB on P.
Hence AP: PB = 3: 5.
Ques. A line segment AB is divided in to the ratio of 4:7, this division has been done by a ray AX which is drawn first such that ∠BAX is an acute angle and then points A1, A2, A3, .... are located at equal distances on the ray AX and the point B is joined to?
(A) A12
(B) A11
(C) A10
(D) A9 (1 mark)
Ans: (B)
Explanation: Here, minimum 4+7 = 11 points are located at an equal – distances on the ray AX and then B is joined to last point, i.e., A11.
Ques. Draw a circle of radius 2 cm. Take two points P and Q at one of its extended diameters and each shall be at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q. (2 marks)
Ans:
Steps of construction:
- Draw a circle with a radius of 2 cm with the center as “O”.
- Draw a diameter of a circle and extend it to 7 cm from the center and mark it as P and Q, as given in the question.
- Draw the perpendicular bisector of the line PO and mark the midpoint as M.
- Then, draw a circle taking M as a center and MO as its radius.
- Now, join the points PA and PB in which the circle with radius MO intersects the circle at points A and B.
- Now, PA and PB are the required tangents.
- Similarly, from the point Q, the tangents could be drawn.
- From that, QC and QD are the required tangents.
Ques. Find the angle between the tangents that are drawn at the endpoints of the two radii in which a pair of tangents to a circle are drawn, which are also inclined to each other at an angle of 60o.
(A) 135o
(B) 90o
(C) 60o
(D) 120o (2 marks)
Ans: D, 1200
Explanation: The angle between them should be 120o because the figure is formed by the intersection point of the pair of tangents, the two endpoints of those two radii at which the tangents are drawn and the center of the circle, is formed to be a quadrilateral. Thus, as per the principle to quadrilaterals, the sum of the opposite angles in the quadrilateral must be 180o.
Previous Year Questions
Ques. Construct a ?ABC in which CA = 6 cm, AB = 5 cm and ∠BAC = 45°. Then construct a triangle whose sides are 3/5 of the corresponding sides of ?ABC. (2019)
Ans. Steps of construction?:
- Draw a line AB = 5cm
- At A draw ∠BAX = 45º
- Cut AC from AX = 6cm
- Join BC to form ????ABC
- Draw AY to form an acute angle with AB
- Draw 5 arcs P?,P?,P?,P? and P? at regular intervals
- Then join BP?
- Draw P?B’ || P?B which meets at AB at B’
- From B’ draw B’C’ || BC meeting AC at C’
????AB’C’????ABC
Hence ????AB’C’ is the triangle that is required
Ques. Construct an equilateral ?ABC with each side 5 cm. Then construct another triangle whose sides are 2/3 times the corresponding sides. Draw two concentric circles of radii 2 cm and 5 cm. Take a point P on the outer circle and construct a pair of tangents PA and PB to the smaller circle. Measure PA. (Outside Delhi 2019)
Ans. Construction:
- Draw a line segment = 5cm
- Construct ∠CBX = 60º and ∠BCX = 60º at B and C
- Cut an arc BX at A keeping B as the center and radius 5 cm.
- Join AC
- We get an equilateral triangle ????ABC
- Below BC form an acute angle ∠CBY
- Along B making three points as B?, B? and B? so that BB?, B?B?, and B?B? are equal.
- Join B?C
- Draw B?D || B?C from B? which meets BC at D
- Draw DE || CA from D which meets AB at E
- Then we get ????EBD that has sides which is ? of the corresponding side of ????ABC
Ques. Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 3/4 of the corresponding sides of the ?ABC. (2018)
Ans.
- Draw a line BC = 6cm
- Construct ∠XBC = 60º
- Draw an arc that intersects XB at A with B as a center and radius = 5cm
- Join AC and we get ????ABC
- Draw an acute angle ∠CBY below B
- Mark 4 parts which are equal in nature on BY as B?, B?, B? and B?
- Join C to B?
- From B draw a parallel line to B?C which intersects BC at C
- From C draw a parallel line to CA which intersects AB at A
- ????A’BC’ is the triangle required which is similar to ????ABC so that BC’ = ¾ BC
Ques. Construct a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then construct another triangle whose sides are 3/4 times the corresponding sides of the ?ABC. (Outside Delhi 2017)
Ans. BC = 7cm, ∠B = 45o, ∠A = 105o
∠C = 180 - (∠B +∠A) = 180 - (45+105) = 180 - 150 = 30
Construction:
- Draw a line BC = 7 cm
- Draw a 45 angle at B and 30 angle at C and they should intersect at A
- At B draw an acute angle
- Divide the angle ray at equal intervals B1, B2, B3 and B4 such that BB1 = B1B2 = B2B3 = B3B4
- Join B3 to C
- Draw a parallel line to B3C from B4 where BC intersects at C’
- Draw a parallel line to CA from C’ where AB intersects at A’
Therefore ????A’BC’ is the triangle required so that ????A’BC’????ABC with A’B = 3/4AB
Ques. Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to a smaller circle from a point on the larger circle. Also measure its length. (2016)
Ans. Given, OD = 3cm and OP = 5cm
PA and PB are the tangents required.
Through measurements PA = PB = 4cm.
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