Question

What is the area of the question mark? The opposite lines are not necessarily parallel.

Answer 1

`200`

Explanation:

Note that the areas are unchanged by shearing, so we can assume without loss of generality that this is an orthodiagonal quad, with vertices at `(a, 0)`, `(0, b)`, `(-c, 0)` and `(0, -d)`. Note that if we scale the `x` and `y` coordinates inversely, then all areas are preserved too. Hence it does not matter what positive value we use for `a` (though I will choose `10` for ease of calculation below).

Remember that the area of a triangle is `1/2 xx "base" xx "height"`

Putting the triangle of area `100` in Q1 and working round anticlockwise, we can put `(a, 0) = (10, 0)`, `(0, b) = (0, 20)`, `(-c, 0) = (-15, 0)`, `(0, -d) = (0, -40)`.

So the last triangle has vertices `(0, 0)`, `(0, -40)` and `(10, 0)`, giving an area of `1/2 xx 40 xx 10 = 200`

Answer 2

`?=200 " units"^2`

Explanation:

`AO:OC=a:b=100:150`,
`=> a:b = 2:3`,
similarly, `?:300=a:b=2:3`,
`=> ?=200 " units"^2`