Question

D. The height of a triangle is increased by 20%.
By how much percent must its base reduced so
that the area is increased by 10%?
एक त्रिभुज की ऊँचाई में 20% की वृद्धि कर दी गई, तो इसके आधार
में कितने प्रतिशत की कमी की जाये कि त्रिभुज का क्षेत्रफल 10%
बढ़े?
​

Answer 1

The Base of the triangle is reduced by 8 %.

Given:

The increase in the height of the triangle = 20 %

The increase in the Area of the triangle = 10 %

To Find:

The per cent decrease in the base of the triangle.

Solution:

Let the initial Height, Base and Area of the triangle be H, B and A.

         $\boldsymbol{\because Area = \frac{1}{2} \times Base \times Height}$

         $\boldsymbol{\therefore A = \frac{1}{2} \times B \times H-Equation(i)}$

→ Since the increase in the height of the triangle = 20 %

∴ The new height of the triangle = (H) + H × (20/100) = 1.2H

→ Since the increase in the Area of the triangle = 10 %

∴ The new Area of the triangle = (A) + A × (10/100) = 1.1A

Let the new Base of the triangle be B'.

         $\boldsymbol{\because Area = \frac{1}{2} \times Base \times Height}$

         $\boldsymbol{\therefore A' = \frac{1}{2} \times B' \times H'}\\\\\boldsymbol{\therefore 1.1A= \frac{1}{2} \times B' \times 1.2H-Equation(ii)}$

→ Dividing Equation (ii) by equation (i):

          $\boldsymbol{\therefore \frac{1.1A}{A}=\frac{(B' \times 1.2H)}{(B\times H)} }\\\\\boldsymbol{\therefore 1.1=1.2\times \frac{B'}{B} }\\\\\boldsymbol{\therefore B'=\frac{1.1}{1.2}B }\\\\\boldsymbol{\therefore B'=0.92B}$

Therefore the new Base of the triangle is equal to 0.92B.

→ The per cent decrease in the base of the triangle:

$\boldsymbol{\because The \:Percent\:Decrease\:in\:the\:Base=\frac{New\:Base-Initial\:Base}{Initial\:Base}\times 100 }$

      $\boldsymbol{\therefore The\:Percent\:Decrease\:in\:the\:Base=\frac{B-0.92B}{B}\times100 }\\\\\boldsymbol{\therefore The\:Percent\:Decrease\:in\:the\:Base=\frac{0.08(B)}{B}\times100 }\\\\\boldsymbol{\therefore The\:Percent\:Decrease\:in\:the\:Base=8 \:\% }$

Therefore the per cent decrease in the Base of the triangle comes out to be equal to 8 %. Hence the Base of the triangle is reduced by 8 %.

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