Construction Chapter Notes - Mathematics (Maths) Class 10

Construction of a triangle similar to a given triangle as per the given scale factor when Scale Factor is more than 1

Q. Construct an isosceles triangle with base 5 cm and equal sides of 6 cm. Then, construct another triangle whose sides are of the corresponding sides of the (4/3)th of first triangle .

Steps of construction:
(i) Draw BC = 5 cm
(ii) With B and C as the centre and radius 6 cm, draw arcs on the same side of BC, intersecting at A.
(iii) Join AB and AC to get the required ΔABC.
(iv) Draw a ray , making an acute angle with BC on the side opposite to the vertex A.
(v) Mark 4 points B₁, B₂, B₃, B₄, along BX such that BB₁ = B₁B₂ = B₂B₃ = B₃B₄
(vi) Join B₃C. Draw a line through B₄ parallel to B₃C, making an angle equal to ∠BB₃C intersecting the extended line segment BC at C´.
(vii) Through point C´, draw a line parallel to CA, intersecting extended BA at A´.
(viii) The resulting ΔA´BC´ is the required triangle.

Division of a line segment in a given ratio 

Q. Draw Line segment PQ=9cm and divide it in the ratio 2:5. Justify your construction.

Steps of construction:
(i) Draw Line Segment PQ=9cm
(ii) Draw a Ray PX, making an acute angle with PQ.
(iii) Mark 7 points A₁ , A₂ , A₃ …A₇ along PX such that PA₁ = A₃A₂ = A₂A₃ = A₃A₄ = A₄A₅ = A₅A₆ = A₆A₇
(iv) Join QA₇
(v) Through the point A₂ , draw a line parallel to A7Q by making an angle equal to ∠PA₇Q at A₂, intersecting PQ at point R. PR:RQ = 2:5

Justification:

We have A₂R || A₇Q

Construction of tangents to a circle

Q. Draw a circle of radius 3 cm. From a point 5 cm away from its centre, construct a pair

Steps of construction:
(i) Draw a circle with centre O and radius 3 cm. Take a point P such that OP = 5 cm, and then join OP.
(ii) Draw the perpendicular bisector of OP. Let M be the mid point of OP.
(iii) With M as the centre and OM as the radius, draw a circle. Let it intersect the previously drawn circle at A and B.
(iv) Joint PA and PB. Therefore, PA and PB are the required tangents. It can be observed that PA=PB=4cm.

Construction of a triangle similar to a given triangle as per the given scale factor when Scale Factor is less than 1

Q. Draw a ΔABC with sides BC = 8 cm, AC = 7 cm, and ÐB = 70°. Then, construct a similar triangle whose sides are (3/5)th of the corresponding sides of the ΔABC. 

Steps of construction:
(i) Draw BC = 8 cm
(ii) At B, draw ∠XBC = 70°
(iii) With C as centre and radius 7 cm, draw an arc intersecting BX at A.
(iv) Join AB, and DABC is thus obtained.
(v) Draw a ray , making an acute angle with BC.
(vi) Mark 5 points, B₁, B₂, B₃, B₄, B₅, along BY such that
(vii) BB₁ = B₁B₂ = B₂B₃ = B₃B₄ = B₄B₅
(viii) Join CB₅
(ix) Through the point B₃, draw a line parallel to B₅C by making an angle equal to ∠BB₅C, intersecting BC at C´.
(x) Through the point C´, draw a line parallel to AC, intersecting BA at A´. Thus, ΔA´BC´ is Required Triangle.

Justification

Using BPT