Challenge Level

ABCD is a square. M is the midpoint of the side AB.

By constructing the lines AC, MC, BD and MD, the blue shaded quadrilateral is formed:

**What fraction of the total area is shaded?**

Below are three different methods for finding the shaded area. Unfortunately, the statements have been muddled up. Can you put them in the correct order?

(A) The shaded area is made up of two congruent triangles,

one of which has vertices $(\frac{1}{3},\frac{2}{3}), (\frac{1}{2},\frac{1}{2}), (\frac{1}{2},1)$.

(B) The line joining $(0,0)$ to $(\frac{1}{2},1)$ has equation $y=2x$

(C) Area of the triangle $= \frac{1}{2}(\frac{1}{2} \times \frac{1}{6}) = \frac{1}{24}$

(D) The line joining $(0,1)$ to $(1,0)$ has equation $y=1-x$.

(E) Therefore the shaded area is $2 \times \frac{1}{24} = \frac{1}{12}$

(F) The point $(a,b)$ is at the intersection of the lines

$y=2x$ and $y=1-x$.

(G) Consider a unit square drawn on a coordinate grid.

(H) The perpendicular height of the triangle is $\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$.

(I) So $a = \frac{1}{3}, b=\frac{2}{3}$.

(J) The line joining $(0,0)$ to $(1,1)$ has equation $y=x$.

(A) As line $AC$ intersects line $MD$ at point $E$,

the two opposite angles $\angle MEF$ and $\angle AED$ are equal.

(B) The line $MF$ is half the length of $AD$.

(C) Line $AD$ is parallel to line $MF$, so $\angle EDA$ and $\angle EMF$ are equal, and $\angle EAD$ and $\angle EFM$ are equal (alternate angles).

(D) Therefore, $\triangle AED$ and $\triangle FEM$ are similar.

(E) Therefore, the line $EH$ is half the length of $PE$.

(F) Let ABCD be a unit square.

(G) Therefore, the shaded area $MEFG = \frac{1}{24} \times 2 = \frac{1}{12}$ sq units.

(H) $PH$ has length $\frac{1}{2}$ units, so $PE$ has length $\frac{1}{3}$ units and $EH$ has length $\frac{1}{6}$ units.

(I) $\triangle MEF$ has area $\frac{1}{2}(\frac{1}{2}\times\frac{1}{6}) = \frac{1}{24}$ sq units.

(A) The area of $\triangle DMC = 2$ sq units.

The area of $\triangle DFC = 1$ sq unit.

Thus the combined area of $\triangle DFE$, $\triangle CFG$ and

shaded area $MEFG$ is $1$ sq unit.

(B) $(EH)^2+(HF)^2=(EF)^2 $

$EH = HF $

$(EH)^2 = \frac{1}{2}(EF)^2$

$EH = \frac{EF}{\sqrt 2}$

(C) Areas of $\triangle DFE$, $\triangle CFG$ and shaded area $MEFG$ are equal

so each must have an area of $\frac{1}{3}$ sq units.

(D) Area of $\triangle MEF = \frac{1}{2}(1 \times EH) = \frac{1}{2}(\frac{EF}{\sqrt 2})$

(E) By Pythagoras, $DF$ has length $\sqrt 2$.

(F) The total area of the square is $4$ sq units, so the shaded area is

$\frac{1}{12}$ the area of the whole square.

(G) Area of $\triangle DFE = \frac{DF \times EF}{2}$

$= \frac{\sqrt 2 \times EF}{2} = \frac{EF}{\sqrt 2}$

(H) So the shaded area $MEFG$ is equal to the area of $\triangle DFE$.

(I) Assume that the sides of the square are each $2$ units long.

Thus, $DJ$ and $FJ$ are each $1$ unit long.