Question

∆PQR ~ ∆LTR. In ∆PQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ∆PQR and ∆LTR, such that `"PQ"/"LT" = 3/4`.

Answer 1

Analysis:

Rough Figure

As shown in fig, let the points R, P, L and RQT be collinear.

∆PQR ~ ∆LTR        ...[Given]

∴ ∠PRQ ≅ ∠LRT … [Corresponding angles of similar triangles]

`"PR"/"LR" = "QR"/"TR" = "PQ"/"LT" = 3/4`.

∴ sides of ∆LTR are longer than corresponding sides of ∆PQR.

∴ the length of side QR will be equal to 3 parts out of 4 equal parts of side TR.

So, if we construct ∆PQR, point T will be on side RQ, at a distance equal to 4 parts form R.

Now, point L is the point of intersection of ray RP and a line through T, parallel to PQ.

∆LTR is the required triangle similar to ∆PQR.

Steps of Construction:       

1. Draw ∆PQR such that PQ = 4.2 cm, QR = 5.4 cm and PR = 4.8 cm, choosing QR = 5.4 cm as base.
2. Draw ray RS making an acute angle with side RQ.
3. Taking convenient distance on the compass, mark 4 points R₁, R₂, R₃, and R₄, such that RR₁ = R₁R₂ = R₂R₃ = R₃R₄.
4. Join R₃Q. Draw line parallel to R3Q through R₄ to intersects ray RQ at T.
5. Draw a line parallel to side PQ through T. Name the point of intersection of this line and ray RP as L.
6. ∆LTR is the required triangle similar to ∆PQR.