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If each sides of a triangle is double then find the ratio of area of the new triangle thus formed and given triangle

Posted by Nithish Nithish 8 years, 3 months ago

CBSE > Class 09 > Mathematics

1 answers

Sahdev Sharma 8 years, 3 months ago

Let a,b,c be the sides of the triangle. Perimeter 2s = a + b + c

Semi-perimeter, s ={tex} a+b+c\over 2{/tex}

Using Heron's formula: Area of the triangle A = {tex}\sqrt {s(s−a)(s−b)(s−c) }{/tex}

Now, if the sides are doubled: 2a, 2b, 2c Let s' be the semi-perimeter.

2s' = 2a + 2b + 2c

s' = a + b + c or s' = 2s

Area of the triangle,

A' = {tex}\sqrt {s′(s′−2a)(s′−2b)(s′−2c) }{/tex}

A' = {tex} \sqrt {(2s)(2s−2a)(2s−2b)(2s−2c)}{/tex}

A'= {tex}\sqrt {16s(s−a)(s−b)(s−c) }{/tex}

A' ={tex} 4\sqrt {s(s−a)(s−b)(s−c) }{/tex}

A' = 4A

A':A = 4:1

Ratio of area of the new triangle and old triangle is 4:1

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