Question

quadrilateral pqrs is square if triangle qrt and triangle pru are similar to each other the prove that area of triangle qrt=1/2area of triangle prq​

Answer 1

Answer:

Given. A square PQRS Also, triangle QRT is an equilateral triangle drawn on side QR and

triangle PRU is an equilateral triangle drawn on diagonal PR

To prove. ar(Qrt) =1/2ar(pqr).

Proof. Both triangle qrt and PRU are equilateral and hence

equiangular and similar.

triangle BCE~ triangle ACF

As the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides, so

ar(qrt)/ar(pru) =qr^2/pr^2 = qr^2/(√2qr) ^2 =1/2 { pr=√2qr}

hence, area of triangle qrt=1/2area of triangle prq

Answer 2

The ratio of areas of two similar triangles is proportional to the squares of the corresponding sides of both the triangles.

Step-by-step explanation:

Given : PQRS is a square

Let side of square is a unit

The diagonal of a square is given by

$H^2=P^2+B^2 \text{ Pythagoras theorem }\\\\PR^2= PQ^2+QR^2\\\\\Rightarrow PR=\sqrt{a^2+a^2} =\sqrt{2} a$

Now

As given Δ QRT ≈Δ PRU

Now as we know

The ratio of areas of two similar triangles is proportional to the squares of the corresponding sides of both the triangles

Therefore

$\dfrac{\text {area of } \triangle QRT}{\text {area of } \triangle PRQ} =( \dfrac{QR}{PR} )^2= (\dfrac{a}{\sqrt{2} a} )^2=\dfrac{1}{2} \\\\\Rightarrow \dfrac{\text {area of } \triangle QRT}{\text {area of } \triangle PRQ}= \dfrac{1}{2} \\\\\Rightarrow \text {area of } \triangle QRT= \dfrac{1}{2} \text {area of } \triangle PRQ$

#Learn more

Prove that the ratio of the areas of two similar triangles is equal to square of the ratio of their corresponding median

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