18. The diagram shows an isosceles right-angled triangle which has a hypotenuse of length $y$. The interior of the triangle is split up into identical squares and congruent isosceles right-angled triangles.
What is the total shaded area inside the triangle?
A $\frac{y^{2}}{2}$
B $\frac{y^{2}}{4}$
C $\frac{y^{2}}{8}$
$\mathrm{D} \frac{y^{2}}{16}$
$\mathrm{E} \frac{y^{2}}{32}$
18. C Let the length of each equal side of the given triangle be $x$.
Then, by Pythagoras' theorem, $x^{2}+x^{2}=y^{2}$. So $x^{2}=\frac{y^{2}}{2}$.
In the triangle, four of the squares are shaded while the unshaded area consists of two squares and four half-squares.
Therefore, half of the area of the triangle is shaded.
Now the area of the triangle is $\frac{1}{2} \times x \times x=\frac{1}{2} x^{2}$.
Therefore the total shaded area of the triangle is $\frac{1}{4} x^{2}=\frac{1}{4} \times \frac{y^{2}}{2}=\frac{y^{2}}{8}$.