Question

∆ABC ~ ∆PBQ. In ∆ABC, AB = 3 cm, ∠B = 90°, BC = 4 cm. Ratio of the corresponding sides of two triangles is 7 : 4. Then construct ∆ABC and ∆PBQ.

Answer 1

Analysis: As shown in the figure,

Let B – A – P and B – C – Q.

∆PBQ ∼ ∆ABC

∴ ∠PQB ≅ ∠ACB   ...[Corresponding angles of similar triangles]

`(PB)/(AB) = (BQ)/(BC) = (PQ)/(AC)`   ...(i) [Corresponding sides of similar triangles]

∴ `(PB)/(AB) = (BQ)/(BC) = (PQ)/(AC) = 7/4`   ...[Given]

∴ Sides of ∆PBQ are longer than corresponding sides of ∆ABC.

∴ If seg BC is divided into 4 equal parts, then seg BQ will be 7 times each part of seg BC.

So, if we construct ∆ABC point Q will be on side BC, at a distance equal to 7 parts from B.

Now, point P is the point of intersection of ray AB and a line through Q, parallel to AC.

∴ ∆PBQ is the required triangle similar to ∆ABC.

Steps of construction:

1. Draw seg BC of length 4 cm.
2. Take ∠B as 90° and draw an arc of 3 cm on it. Name the point as A.
3. Join seg AC to obtain ∆ABC.
4. Draw ray BX such that ∠CBX is an acute angle.
5. Locate points B₁, B₂, B₃, B₄, B₅, B₆, B₇ on ray BX such that, BB₁ = B₁B₂ = B₂B₃ = B₃B₄ = B₄B₅ = B₅B₆ = B₆B₇.
6. Join point C and B₄.
7. Through point, B₇ draw a line parallel to seg CB₄ which intersects seg BC at point Q.
8. Draw a line parallel to AC through Q to intersect line AB at point P.
   ∆PBQ is the required triangle similar to ∆ABC.