video explanation

text explanation

think about what we know.  we know that both a and b are positive, because point a is in the first quadrant.  

point b is at (4a, b).  it has the same y-coordinate as point a, so the line segment ab must be horizontal.   it is is a horizontal line from x = a to x = 4a.  to find distance along a horizontal or vertical line in the x-y plane, we simply subtract.   

length of ab = 4a – a = 3a 

that's the length of the base of the triangle.  

now consider point c.  point c = (2a, 5b) has an x-value that is between a & b, and it's above them.  a & b are at a height of y = b, and c is at a height of y = 5b.  to find the distance, again, we simply subtract: 

height = 5b – b = 4b  

that's the height of the triangle.  

we have base and height, so we can find the area.   

area = (3a)(4b) = (3a)(2b) = 6ab

answer = (e) 

faq: how do i know where to put those other points? there are no numbers! only variables.

a: great question! and always, if you have trouble with variables, choose some numbers to replace the variables. always choose easy numbers to work with too.

in this case, i would choose these numbers:

a = 1

b = 1

now that makes drawing out triangle a lot easier.

point a is (1,1)

point b is (4,1)

point c is (2, 5)

with these numbers, it is much easier to calculate the base and height of the triangle. once you have the base and height and calculate the area, turn to the answer choices and plug in one for a and b to find the answer choice that matches your calculation.