Construction Of A Triangle Similar To A Given Triangle

SCALE FACTOR:

The ratio of the sides of the triangle to be constructed with the corresponding sides of the given triangle is known as their scale factor.

Steps of Construction when m < n:

(i) Construct the given triangle ABC by using the given data.

(ii) Take any one of the three side of the given triangle as base. Let AB be the base of the given triangle.

(iii) At one end, say A, of base AB. Construct an acute angle ∠BAX below the base AB.

(iv) Along AX mark of n points A₁, A₂,A₃,.....An such that AA₁ = A₁A₂ = ..... = A_n-1A_n.

(v) Join An B.

(vi) Draw AmB’ parallel to An B which meets AB at B’.

(vii) From B’ draw B’ C’|| CB meeting AC at C’.

Triangle AB’C’ is the required triangle each of whose side is (m/n)^th of the corresponding side of Δ ABC.

question 1. Construction a ΔABC in which AB = 5 cm, BC = 6 cm and AC = 7 cm. Now, construct a triangle similar to ΔABC such that each of its side is two-third of the corresponding side of ΔABC.

Solution: Steps of Construction

(i) Draw a line segment AB = 5 cm.

(ii) With A as centre and radius AC = 7 cm, draw an arc.

(iii) With B as centre and BC= 6 cm, draw another arc, intersecting the arc draw in step (ii) at C.

(iv) Join AC and BC to obtain ΔABC.

(v) Below AB, make an acute angle ∠BAX.

(vi) Along AX, mark off three points (greater of 2 and 3 in 2/3) A₁,A₂,A₃ such that AA₁ = A₁A₂ = A₂A₃.

(vi) Join A₃B.

(viii) Draw A₂B’ || A3B, meeting AB at B’.

(iv) From B’, draw B’C’ || BC, meeting AC at C’.

AB’C’ is the required triangle, each of the whose sides is two-third of the corresponding sides of ΔABC.

Steps of Construction when m > n:

(i) Construct the given triangle by using the given data.

(ii) Take any of the three sides of the given triangle and consider it as the base. Let AB be the base of the given triangle.

(iii) At one end, say A, of base AB construct an acute angle ∠BAX below base AB i.e. on the composite side of the vertex C.

(iv) Along AX, mark-off m (large of m and n) points A₁, A₂,.....Am on AX such that AA₁ = A₁A₂ = .... A_m-1 A_m.

(v) Join An to B and draw a line through Am parallel to An B, intersecting the extended line segment AB at B’.

(vi) Draw a line through B’ parallel to BC intersecting the extended line segment AC at C’.

(vii) ΔAB’C’ so obtained is the required triangle.

question 1. Draw a triangle ABC with side BC = 7 cm, ∠B = 450, ∠A = 1500 Construct a triangle whose side are (4/3) times the corresponding side of ΔABC.

Solution: In order to construct ΔABC, follow the following steps :

(i) Draw BC = 7 cm.

(ii) At B construct ∠CBX = 450 and at C construct ∠BCY = 1800 - (450 + 1050) = 300

Suppose BC and CY intersect at A. ΔABC so obtained is the given triangle.

(iii) Construct an acute angle ∠CBZ at B on opposite side of vertex A of ΔABC.

(iv) Mark-off four (greater of 4 and 3 in 4/3) points, B₁,B₂,B₃,B₄ on BZ such that BB₂ - B₁B₂ = V₂B₃ = B₃B₄.

(v) Join B₃ ( the third point) to C and draw a line through B₄ parallel to B₃C, intersecting the extended line segment BC at C’.

(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