a square is split into four triangles, and then three of the four triangles are shaded as shown. If the areas of the shaded triangles are 3,4, and 6, as shown, what is the area of the unshaded triangle?

For the Triangle with area 6, we then have ((x-a)(x-b))/2 = 6. Expanding gives us x^2-bx-ax+ab=12. Using the equation above and plugging in values where bx = 6, ax=8, a = 8/x, and b = 6/x. We get

x^2-6-8+48/x^2 = 12.

Simplified gives us,

x^4-26x^2+48.

Factored gives us,

((x^2-2)(x^2-24).

Since the area cannot be any negative values and bigger than 6 (since there is already a triangle with area 6), we then have x^2 = 24.

Therefore a of unshaded region = 24-6-4-3= 11

MrRoyal MrRoyal · Answer 2 · 2021-10-11T07:18:27+02:00

The area of a shape is the amount of space it occupies.

The area of the unshaded triangle is: [tex]\mathbf{x^2 - 13}[/tex]

The area of three triangles are:

[tex]\mathbf{A_1 = 3}[/tex]

[tex]\mathbf{A_2 = 4}[/tex]

[tex]\mathbf{A_3 = 6}[/tex]

Assume the side length of the square is x.

So, the area of the square is:

[tex]\mathbf{Area = x \times x}[/tex]

[tex]\mathbf{Area = x^2}[/tex]

The area of the unshaded triangle (A) is the sum of the areas of the three triangles, subtracted from the area of the square

So, we have:

[tex]\mathbf{A = Area - (A_1 + A_2 + A_3)}[/tex]

Substitute known values

[tex]\mathbf{A = x^2 - (3+4+6)}[/tex]

[tex]\mathbf{A = x^2 - 13}[/tex]

Hence, the area of the unshaded triangle is: [tex]\mathbf{x^2 - 13}[/tex]