Question

Construct ∆PYQ such that, PY = 6.3 cm, YQ = 7.2 cm, PQ = 5.8 cm. If \[\frac{YZ}{YQ} = \frac{6}{5},\] 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]

\[\frac{XY}{PY} = \frac{YZ}{YQ} = \frac{XZ}{PQ}\]          …(i)   [Corresponding sides of similar triangles]

But, \[\frac{YZ}{YQ} = \frac{6}{5}\]        …[From (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:

1. Draw ∆ PYQ of given measure. Draw ray YT making an acute angle with side YQ.
2. Taking convenient distance on compass, mark 6 points Y₁, Y₂, Y₃, Y₄, Y₅ and Y₆ such that YY₁ = Y₁Y₂ = Y₂Y₃ = Y₃Y₄ = Y₄Y₅ = Y₅Y₆.
3. Join Y₅Q. Draw line parallel to Y₅Q through Y₆ to intersects ray YQ at Z.
4. 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.