Question 1006494
if the triangles are similar, then all corresponding parts are proportional which means the ratios of the corresponding parts are the same.


the altitude of a triangle counts as a corresponding part.


let the area of triangle 1 be 2/3 * b * h
b is the base
h is the height.


let the area of triangle 2 be 2/3 * xb * xh


x is the common ratio.


A1 = area of triangle 1 = 1/2 * b * h


A2 = area of triangle 2 = 1/2 * xb * xh


the ratio of the area of A2 to the area of A1 is (1/2 * xb * xh) / (1/2 * b * h)


since xb * xh is the same as x^2 * b * h, the ratio of A2 to A1 becomes:


A2 / A1 = (1/2 * x^2 * b * h) / (1/2 * b * h)


the (1/2 * b * h) in the numerator cancels out with the (1/2 * b * h) in the denominator and you are left with:


A2 / A1 = x^2


x is the common ratio of the corresponding parts.


x^2 is the square of the ratio of the coresponding sides.