Question

In the triangle PQR below,
- S and T are 2 points on the sides RP and RQ respectively such that ST is parallel to PQ.
- The ratio of RT to TQ is 1:2.
The area of ΔRST = 100 sq. units.
What is the area of PQTS?

Answer 1

800 $cm^{2}$

Answer 2

Area of PQTS is 800 sq units if in ΔPQR , ST || PQ and RT:TQ = 1 : 2 and area of ΔRST = 100 sq. units.

Given:

• ΔPQR
• S and T are points on Sides RP and RQ respectively
• ST || PQ
• The ratio of RT to TQ is 1:2
• The area of ΔRST = 100 sq. units

To Find:

• Area of PQTS

Solution:

Corresponding  angles : A pair of angles that occupy the same relative position at each intersection by a transversal line

Corresponding angles formed by transversal line  with two parallel lines are congruent. ( Equal in Measure)

Step 1:

Show similarity of  ΔRPQ and ΔRST

∠R = ∠R  ( common)

∠P = ∠S   Corresponding angles as ST || PQ

∠Q = ∠T  Corresponding angles as ST || PQ  

=> ΔRPQ ~ ΔRST  (Using AAA similarity)

Step 2:

Find  Ratio of corresponding sides of  similar trianglesΔRPQ and ΔRST

RQ/RT  = (RT + TQ)/RT

= 1 + TQ/RT

(RT : TQ = 1 : 2   => TQ = 2 RT => TQ/RT = 2)

= 1 +  2

= 3

Step 3:

Use formula that Ratio of area of similar triangle is square of ratio of corresponding side of similar triangles

Ar Δ  RQP / Ar Δ  RST  = (RQ/RT)²

=> Ar Δ  RQP / 100  = 3²

=> Ar Δ  RQP = 900 sq units

Step 4:

Subtract Ar Δ  RST  from Ar Δ  RQP to find area of  PQTS

Ar PQTS = Ar Δ  RQP - Ar Δ  RST

=> Ar PQTS = 900 - 100

=> Ar PQTS = 800 sq units

Area of PQTS is 800 sq units

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