Find the percentage increase in the area of a triangle if each side is increased to 55 times.
Answer:
2400%2400%
- Consider a triangle QRSQRS with sides a,ba,b and c.c.
Let S=a+b+c2S=a+b+c2
Area of triangle 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 55 times, we get a new triangle XYZ.XYZ.
XYZXYZ has sides 5a,5b5a,5b and 5c.5c.
By Heron's formula,
Area of new triangle =√S1(S1−5a)(S1−5b)(S1−5c)=√S1(S1−5a)(S1−5b)(S1−5c)
Where S1=5a+5b+5c2=5×a+b+c2S1=5a+5b+5c2=5×a+b+c2
Area of XYZ=√5S(5S−5a)(5S−5b)(5S−5c)XYZ=√5S(5S−5a)(5S−5b)(5S−5c)
=√54S(S−a)(S−b)(S−c)=52×A1=25A1 - Percentage increase in the area of the triangle, = Area of Triangle XYZ − Area of Triangle QRS Area of Triangle QRS×100=25A1−A1A1×100=24A1A1×100=2400
- This means the area of the triangle, A1 is increased by 2400%.