Question

PQR is an isosceles triangle right angled at Q. Two equilateral triangles PRS and PQT are constructed on the sides PR and PQ respectively.Find the ratio of the areas of triangle PQT and triangle PRS

Answer 1

Define x:

Let the length of PQ be x

The length of QR = x (isosceles triangle)

Find the length PR:

a² + b² = c²

x² + x² = c²

2x² = c²

c √(2x²)

c = √2 x units

Find the ratio of area of ΔPQT : area of ΔPRS:

Both  ΔPQT and  ΔPRS are equilateral triangle

⇒ They are similar triangles.

$\dfrac{\text{Area of PQT} }{\text{Area of PRS} }= \bigg (\dfrac{PQ}{PR} \bigg)^2$

$\dfrac{\text{Area of PQT} }{\text{Area of PRS} }= \bigg (\dfrac{x}{\sqrt{2} x} \bigg)^2$

$\dfrac{\text{Area of PQT} }{\text{Area of PRS} }= \dfrac{x^2}{2 x^2}$

$\dfrac{\text{Area of PQT} }{\text{Area of PRS} }= \dfrac{1}{2}$

Answer: The ratio is 1 : 2

Answer 2

hey!

here is answer

let the. length pq be x

the length of. qr= x(isosceles triangle)

length of. pr

a^2+b^2=c^2


x^2+x^2=c^2

2x^2=c^2

$c = \sqrt{2 {x}^{2} }$


$c = \sqrt{2x}$


ratio of area of triangle PQT and PRS

next is in pic


hope it helps

thanks