Answer:
a. x²
b. 144 cm²
Explanation:
You want to know an expression for the area of the four white triangles in the given figure, and you want to know that area when x = 12 cm.
a. Triangle area
If you draw horizontal and vertical lines through the middle of the figure, you can see that for each white triangle there is a matching shaded triangle. That is, the shaded area is equal to the white area.
The shaded area is the square of the length of the side of the shaded square, so is ...
shaded area = x²
The white area is the same, so is ...
white area = x²
b. Area for x=12 cm
Using x = 12 cm, the formula from part (a) tells us ...
white area = (12 cm)² = 144 cm²
The total area of the four white triangles is 144 cm².
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Additional comment
The relation between the side lengths of the shaded area and the side length of the total area is ...
2x/√2 = x+6
This has one solution: x = 6/(√2 -1) ≈ 14.485281 (cm).
The figure cannot be drawn to scale for any other value of x, so it is impossible for x=12 cm, unless the outside length is (x+4.97056...)
If you take the white triangles to be isosceles right triangles, each will have an area of A = 1/2bh = 1/2(x/√2)(x/√2) = x²/4. The four of them will have an area of 4(x²/4) = x². If you take the white area to be the overall area less the area of a square of side length x, then the area of the four triangles will be (x+6)² -x² = 12x+36. You will notice that these expressions give different values of the area of the white triangles when x=12. (See the previous comment about the value of x.)
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