Question

The perimeter of two similar triangles ABC and triangle PQR are 35 cm and 45 cm respectively, then the ratio of the areas of the two triangle is

Answer 1

Method 1)

Given:-

• ∆ABC and ∆PQR are similar triangles.
• Perimeter of ∆ABC is 35 cm and perimeter of ∆PQR is 45 cm.

Find:-

The ratio of the area of ∆ABC and ∆PQR.

Solution:-

Since ∆ABC and ∆PQR are similar.

Then, the ratio of their corresponding sides is equal.

$\implies\:\sf{\dfrac{AB}{PQ}\:=\:\dfrac{BC}{QR}\:=\:\dfrac{AC}{PR}}$

Let all of the corresponding sides = a

Now,

The perimeter of ∆ABC = Sum of it's all sides

⇒ AB + BC + AC

⇒ aPQ + aQR + aPR

⇒ a(Perimeter of ∆PQR)

So,

$\sf{\dfrac{Perimeter\:of\:\triangle\:ABC}{Perimter\:of\:\triangle\:PQR}\:=\:a}$

The ratio of the area of two similar triangles is equal to the ratio of the square of it's corresponding side.

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\: \bigg (\dfrac{AB}{PQ}\bigg)^{2}}$

But,

$\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\:\dfrac{Perimeter\:of\:\triangle\:ABC}{Perimeter\:of\:\triangle\:PQR}}$

So,

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\: \bigg (\dfrac{Perimeter\:of\:\triangle\:ABC}{Perimeter\:of\:\triangle\:PQR}\bigg)^{2}}$

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\: \bigg (\dfrac{35}{45}\bigg)^{2}}$

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\: \bigg (\dfrac{7}{9}\bigg)^{2}}$

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\:\dfrac{49}{81}}$

•°• Ratio of area of two similar triangles i.e. ∆ABC and ∆PQR is 49:81

Method 2)

The perimeter of two similar triangles i.e. ∆ABC and ∆PQR are 35 cm and 45 cm.

The ratio of the area of two similar triangles is equal to the ratio of the square of it's corresponding side.

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\: \bigg (\dfrac{Perimeter\:of\:\triangle\:ABC}{Perimeter\:of\:\triangle\:PQR}\bigg)^{2}}$

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\: \bigg (\dfrac{35}{45}\bigg)^{2}}$

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\: \bigg (\dfrac{7}{9}\bigg)^{2}}$

⇒ $\sf{\dfrac{Area\:of\:\triangle\:ABC}{Area\:of\:\triangle\:PQR}\:=\:\dfrac{49}{81}}$

•°• Ratio of area of two similar triangles i.e. ∆ABC and ∆PQR is 49:81