Areas All Sliced Up

Geometry Level 1

Given that the above diagram is a square and the area of $Y=2016$ , find the area of $X+Z$ .

The answer is 2016.

3 solutions

Shaun Leong
Jan 28, 2016

$2016=\frac {a+b}{2}(a+2a+b)$ $=\frac {3a^2+4ab+b^2}{2}$ $=\frac {a}{2}(3a+4b)+ \frac {b^2}{2}$ $=X+Z$ Hence $X+Z=\boxed {2016}$

if you can plz explain how did you get (a+2a+b) as the sum of two parallel sides of trapezium Y

Rupesh Gesota - 5 years, 4 months ago

The line running from the top left corner to the bottom right corner is a diagonal of a square.

Consider the white top left triangle. Its angles are 90, 45 and 45 degrees, so it is isoceles. Hence the length of top side of Y is a.

Similarly, the black bottom right triangle is isoceles as well and has side b. Hence the length of the bottom parallel side of Y is $2a+2b-b=2a+b$ .

Shaun Leong - 5 years, 4 months ago

Yes... good observation... thanks for the prompt response.... The way I solved this is: Let the middle white area be M... The middle horizontal slice is same as the diagonally cut slice.. (both are halves).... So X + M + Z = Y + M .... Hence, Y = X + Z

Rupesh Gesota - 5 years, 4 months ago

@Rupesh Gesota – Oh! I never thought about it that way. My first instinct was to bash. Thanks for sharing!

Shaun Leong - 5 years, 4 months ago

Chris Smith
Jan 28, 2016

I used logic. The area of y was from a segment half the height of the square

We know it's a square because a + (a+b) + b = 2 (a+b)

As the areas of x and z are on the opposite side of the diagonal, then combined they must also equal the area of y in this instance

Therefore

Y = 2016 = X + Z

汶良林
Jan 30, 2016

Good visual solution!

Lee Care Gene - 5 years, 4 months ago

Areas All Sliced Up

The answer is 2016.

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