Lab Manual: Ratio of Areas of Two Similar Triangles - Class 10 PDF Download

Objective

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To verify “The ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides” by performing an activity.

Prerequisite Knowledge
(i) Concept of parallel lines.
(ii) Division of a line in a given ratio.

Materials Required
Chart paper, construction box, coloured pens, a pair of scissors, fevicol. 

Procedure

• Take a chart paper and cut a ∆ABC with AB = 6cm, BC = 6cm, CA = 6cm.
• Mark 5 points P₁, P₂, …. , P₅ at a distance of 1cm each on side AB and Q₁, Q₂,…..,Q₅ at a distance of 1cm each on side AC as shown in fig.(i).
  
• Join P₁Q₁, P₂Q₂, …………, P₅Q₅ as shown in fig. (ii).
  
• Draw lines parallel to AC from P₁, P₂, P₅ and also draw lines parallel to AB from the points Q₁, Q₂, ……., Q₅ as shown in fig. (iii).

• Thus ∆ABC is divided into 36 smaller triangles and all are similar to each other and of equal area.
• Construct a ∆PQR with PQ = 1/2 of AB, PR = 1/2 of AC and QR = 1/2 of BC i.e. 3cm each on another chart paper.
• Mark D₁, D₂ and E₁, E₂ on sides PQ and PR respectively.
• Repeat steps 3 and 4.
• Thus ∆PQR is divided into 9 smaller similar triangles equal in area.
  

Observation

• area of ∆ABC = area of 36 smaller ∆’s
• area of ∆PQR = area of 9 smaller ∆’s
• PQ/AB = 3/6 = 1/2 = PR/AC
• Area of ΔPQR / Area of ΔABC = PQ² / AB²
  = [9 smaller Δ’s/36 smaller Δ’s = 14 = (1/2)²
  (because ΔABC ∼ ΔPQR)

Result
Thus it is verified that the ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

Learning Outcome
Concept of area theorem is clear to the students through this activity.

Activity Time
Take isosceles similar triangles and scalene similar triangles and try to verify this activity. Here isosceles triangles, ∆ABC ~ ∆PQR. Scalene triangle ∆DEF ~ ∆KLM