Question

if AR =12m and AE= 15m, what is the ratio of the area of triangle ARH to the total area of triangle AES

Answer 1

Hi there!
So to start off, you first have to know that the ratio of the sides of the triangles are the ratio of the areas of the two triangles.
AR belongs to triangle ARH and AE belongs to triangle AES.
Thus, the ratio of the sides of the two triangles are AR:AE or 12:15
That can be reduced to 4:5, which is also the ratio of the areas of the triangle. 

Answer:  4:5

Hope this helps :)

Answer 2

Answer:

$\frac{16}{25}$

Step-by-step explanation:

we know that

If two figures are similar

then

the ratio of their areas is equal to the scale factor squared

In this problem i will assume that triangle ARH is similar to triangle AES

therefore

the scale factor is equal to

$\frac{12}{15}=\frac{4}{5}$

and the ratio  of the area of triangle ARH to the total area of triangle AES is equal to the scale factor squared

$(\frac{4}{5})^{2} =\frac{16}{25}$