16. Two congruent isosceles right-angled triangles each have squares inscribed in them as shown. The square $\mathrm{P}$ has an area of $45 \mathrm{~cm}^{2}$.
What is the area, in $\mathrm{cm}^{2}$, of the square $\mathrm{R}$ ?
A 40
B 42
C 45
D 48
E 50
SolUtion
A

The diagram on the left shows a dissection of the triangle and square $\mathrm{P}$ into 4 congruent triangles. They are congruent because they each have the same angles $\left(90^{\circ}, 45^{\circ}, 45^{\circ}\right.$ ) and have one side whose length is equal to the side of the square. Since $P$ has area $45 \mathrm{~cm}^{2}$, each of the small triangles has area $22.5 \mathrm{~cm}^{2}$. Hence the large triangle has area $90 \mathrm{~cm}^{2}$.
The diagram on the right shows a dissection of the triangle and square $\mathrm{R}$ into 9 congruent triangles. They are congruent because they each have the same angles $(90,45,45)$ and have one side whose length is equal to half a diagonal of the square. Each of these nine triangles has area, in $\mathrm{cm}^{2}$, of $90 \div 9=10 \mathrm{~cm}^{2}$. Hence the area of square $\mathrm{R}$ is $4 \times 10=40 \mathrm{~cm}^{2}$.