Find the percentage increase in the area of a triangle if each side is increased by nn times.
Answer:
100(n2−1)%100(n2−1)%
- Consider a △QRS△QRS with sides a,ba,b and c.c. Let S=a+b+c2S=a+b+c2.
Area of △QRS,A1=√S(S−a)(S−b)(S−c)△QRS,A1=√S(S−a)(S−b)(S−c) - Increasing the side of each side by nn times, we get a new △XYZ△XYZ.
△XYZ△XYZ has sides na,nbna,nb and ncnc. - By Heron's formula:
Area of new triangle =√S1(S1−na)(S1−nb)(S1−nc)=√S1(S1−na)(S1−nb)(S1−nc)
Where, S1=na+nb+nc2=n×a+b+c2S1=na+nb+nc2=n×a+b+c2
Area of △XYZ=√nS(nS−na)(nS−nb)(nS−nc)=√n4S(S−a)(S−b)(S−c))=n2×A1 - Increase in area =n2A1−A1
% Increase in area =A1(n2−1)A1100= 100(n2−1)%.