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

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2 Answers
George C.
Apr 11, 2018

#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#

CW
Apr 12, 2018

#?=200 " units"^2#

Explanation:

enter image source here
#AO:OC=a:b=100:150#,
#=> a:b = 2:3#,
similarly, #?:300=a:b=2:3#,
#=> ?=200 " units"^2#

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