Three problems about fitting congruent shapes together:

(A) Rectangle $ABCD$ has sides $a, b$, and rectangle $PQRS$ has sides $2a, 2b$. Show that $PQRS$ has 4 times the area of $ABCD$. Does this mean that 4 copies of rectangle $ABCD$ will fit into rectangle $PQRS$? Check and see!

(B) $\Delta ABC$ has sides $a, b, c$, and $\Delta PQR$ has sides $2a, 2b, 2c$. Show that $\Delta PQR$ has 4 times the area of $\Delta ABC$. Does this mean that 4 copies of $\Delta ABC$ will fit into $\Delta PQR$? Check and see!

(C) $\Delta ABC$ has sides $a, b, c$, and $\Delta PQR$ has sides $3a, 3b, 3c$. Show that $\Delta PQR$ has 9 times the area of $\Delta ABC$. Does this mean that 9 copies of $\Delta ABC$ will fit into $\Delta PQR$? Check and see!

(A) Area of rectangle ABCD $= a \times b = ab$

Area of rectangle PQRS $= 2a \times 2b = 4ab$

So, rectangle PQRS has 4 times the area of rectangle ABCD.

$$A = lb$$

Now check fitting:

One side of PQRS is $2a$, so two rectangles of side $a$ fit along it.

Other side is $2b$, so two rectangles of side $b$ fit along it.

Hence, $2 \times 2 = 4$ copies of rectangle ABCD fit exactly into PQRS.

Yes, 4 copies will fit exactly.

(B) If all sides of a triangle are doubled, the triangles are similar with scale factor 2.

Area changes by square of scale factor:

$$2^2 = 4$$

So,

Area of $\Delta PQR = 4 \times$ Area of $\Delta ABC$

Area scale factor $= k^2$

Now check fitting:

A triangle with doubled sides can be divided into 4 congruent triangles equal to the original triangle.

Yes, 4 copies of $\Delta ABC$ fit exactly into $\Delta PQR$.

(C) Scale factor $= 3$

Area scale factor $= 3^2 = 9$

Therefore, Area of $\Delta PQR = 9 \times$ Area of $\Delta ABC$

If sides are multiplied by 3, area is multiplied by $3^2 = 9$

A triangle with sides tripled can be divided into 9 congruent triangles equal to the original triangle.

Yes, 9 copies of $\Delta ABC$ fit exactly into $\Delta PQR$.