Finding Value of Expression in a Triangle

Question

Kindly advise how do we solve this type of question. So far we know that it is a right-angled triangle. Thank you

Let ABC be a triangle with area 30. Let D be any point in its interior and let e, f and g denote the distances from D to the sides of the triangle. What is the value of the expression $5e+12f+13g$?

Answer 1

The area of the big triangle is the sum of $$13\cdot g+(12-e)f+2ef+(5-f)e=60$$ so $$12g+5e+12f=60$$

Answer 2

$5e$ is twice the area of $\triangle ACD$,so does the others.Therefore $5e+12f+13g$ is twice the area of $\triangle ABC$,which is 60.

Answer 3

How do you "know" $\triangle ABC$ is a right triangle? Is it a given?

If you're actually told that it's a $5-12-13$ right triangle as depicted, then just decompose the triangle into the smaller triangles $\triangle ADC, \triangle ADB$ and $\triangle CDB$, each of which has the respective area $\frac 12(5e), \frac 12 (13g)$ and $\frac 12(12f)$. The sum of these is obviously $30$, so you get $5e + 12f + 13g = 30$.

(To answer the confusion in the comment - the smaller triangles are not right triangles, nor do they have to be. The area of any triangle is half times the base times the height, and that's what's being applied here. For instance, in $\triangle ADC$, the height is $e$ while the base is $5$).