Here are the solutions to questions 21, 22, and 23. Question 21: Step 1: Identify corresponding sides. Given that the triangles are similar, the ratio of corresponding sides is equal. Side $AB$ corresponds to side $DE$. Side $BC$ corresponds to side $EF$. Step 2: Set up the proportion and substitute known values. $$ \frac{AB}{DE} = \frac{BC}{EF} $$ $$ \frac{6\, \text{cm}}{DE} = \frac{8\, \text{cm}}{12\, \text{cm}} $$ Step 3: Solve for $DE$. $$ 6 \times 12 = 8 \times DE $$ $$ 72 = 8 \times DE $$ $$ DE = \frac{72}{8} $$ $$ DE = 9\, \text{cm} $$ The length of side $DE$ is $\boxed{\text{9 cm}}$. Question 22: Step 1: Understand the given ratio. The ratio of corresponding sides of the smaller triangle to the larger triangle is $3:5$. Let $S_s$ be the side of the smaller triangle and $S_l$ be the corresponding side of the larger triangle. $$ \frac{S_s}{S_l} = \frac{3}{5} $$ Step 2: Substitute the given side of the smaller triangle. Given $S_s = 9\, \text{cm}$. $$ \frac{9\, \text{cm}}{S_l} = \frac{3}{5} $$ Step 3: Solve for $S_l$. $$ 9 \times 5 = 3 \times S_l $$ $$ 45 = 3 \times S_l $$ $$ S_l = \frac{45}{3} $$ $$ S_l = 15\, \text{cm} $$ The corresponding side of the larger triangle is $\boxed{\text{15 cm}}$. Question 23: Step 1: Identify corresponding sides and the given ratio. The triangles $\triangle XYZ$ and $\triangle DEF$ are similar. Side $XY$ corresponds to side $DE$. The ratio of corresponding sides is $3:12$. Assuming the first number corresponds to the smaller triangle and the second to the larger. $$ \frac{XY}{DE} = \frac{3}{12} $$ Step 2: Substitute the given length of $XY$. Given $XY = 8\, \text{cm}$. $$ \frac{8\, \text{cm}}{DE} = \frac{3}{12} $$ Step 3: Solve for $DE$. $$ 8 \times 12 = 3 \times DE $$ $$ 96 = 3 \times DE $$ $$ DE = \frac{96}{3} $$ $$ DE = 32\, \text{cm} $$ The length of side $DE$ is $\boxed{\text{32 cm}}$. That's 2 down. 3 left today — send the next one.