Question

Construct ∆PYQ such that, PY = 6.3 cm, YQ = 7.2 cm, PQ = 5.8 cm. If = YZ/YQ = 6/5  then construct ∆XYZ similar to ∆PYQ.

Answer 1

                  Rough Figure

Analysis: As shown in the figure, 

Let Y – Q – Z and Y – P – X. ∆XYZ ~ ∆PYQ …[Given] 

∴ ∠XYZ ≅ ∠PYQ …[Corresponding angles of similar triangles]

XY/PY = YZ/YQ = XZ/PQ  ...(i) [Corresponding sides of similar triangle ] 

But, YZ/YQ = 6/5,  ...(ii) [Given]

∴ YZ/PY = YZ/YQ = XZ/PQ = 6/5  ...[Form (i) and (ii)]

∴ sides of ∆XYZ are longer than corresponding sides of ∆PYQ. 

∴ If seg YQ is divided into 5 equal parts, then seg YZ will be 6 times each part of seg YQ. 

So, if we construct ∆PYQ, point Z will be on side YQ, at a distance equal to 6 parts from Y. 

Now, point X is the point of intersection of ray YP and a line through Z, parallel to PQ. 

∆XYZ is the required triangle similar to ∆PYQ.

Steps of construction: 

i. Draw ∆ PYQ of given measure. Draw ray YT making an acute angle with side YQ.

ii. Taking convenient distance on compass, mark 6 points Y₁ , Y₂ , Y₃ , Y₄ , Y₅ and Y₆ such that 

YY₁ = Y₁ Y₂ = Y₂ Y₃ = Y₃ Y₄ = Y₄ Y₅ = Y₅ Y₆ . 

iii. Join Y₅ Q. Draw line parallel to Y₆ Q through Y to intersects ray YQ at Z. 

iv. Draw a line parallel to side PQ through Z. Name the point of intersection of this line and ray YP as X. 

∆XYZ is the required triangle similar to ∆PYQ.