Basic trapezium or not

Geometry Level 3

a and b are 2 parallel side of the trapezium . note that the figure is not to scale and the outer diagram is a rectangle. Imagine a rectangle with sides a,b. find a quarter of the square of its perimetre minus 4 times its area.

256 64 1024 512 16 2048 32 128

Aareyan Manzoor by Aareyan Manzoor , 18, USA

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1 solution

Aareyan Manzoor
Nov 9, 2015

we see that a + 4 + 12 = b ⟹ a − b = − 16 a+4+12=b\Longrightarrow a-b=-16 if the rectangle has sides a,b: { p e r i m e t r e = 2 a + 2 b a r e a = a b \begin{cases} perimetre = 2a+2b\\area=ab\end{cases} . so p e r i m e t r e 2 4 = ( 2 a + 2 b ) 2 4 = 4 a 2 + 8 a b + 4 b 2 4 = a 2 + 2 a b + b 2 \dfrac{perimetre^2}{4}=\dfrac{(2a+2b)^2}{4}=\dfrac{4a^2+8ab+4b^2}{4}=a^2+2ab+b^2 and 4 times area= 14ab. when subtracted: a 2 + b 2 + 2 a b − 4 a b = ( a ) 2 + ( b ) 2 − 2 ( a ) ( b ) = ( a − b ) 2 a^2+b^2+2ab-4ab=(a)^2+(b)^2-2(a)(b)=(a-b)^2 we know what is a-b. put it in and ( − 16 ) 2 = 256 (-16)^2=256

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Can you also explain what 3, 4, 12 are supposed to represent in the image? At least state that those are length of the line segments. Also, which line segments would they represent? The line segment of 12 seems almost equal to the line segment of 4, so that should be made clear.

Is the outline of the diagram a rectangle? Or a parallelogram? Given that you've only drawn 1 pair of parallel lines, I can only conclude that the outline is only another trapezium, so we do not necessarily have a + 16 = b a + 16 = b .

As not all of your thoughts have made it down onto the screen, I'm having trouble understanding what you intend to say.

Calvin Lin Staff - 5 years, 7 months ago

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is the problem more explicable now sir.

Aareyan Manzoor - 5 years, 7 months ago

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i should note down all points in future problem.

Aareyan Manzoor - 5 years, 7 months ago

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@Aareyan Manzoor – Thanks. This is much better.

Yes, by writing up the solution, you should become aware of all of the points that you used in order to solve the problem. This allows you to identify places where you had to make an assumption, because it was not clearly stated in the problem.

Calvin Lin Staff - 5 years, 7 months ago

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