Let's solve for \( x \) and \( y \) using the properties of similar triangles. ### Step 1: Set up the proportions Given that the triangles are similar, corresponding sides are proportional: For triangle \( \triangle ABC \) (big triangle): - \( AB = 40 \) - \( AC = 20 \) - \( BC = 50 \) For triangle \( \triangle abc \) (small triangle): - \( ab = x \) - \( ac = 14 \) - \( bc = y \) **Corresponding sides:** - \( AB \) corresponds to \( ab \) - \( AC \) corresponds to \( ac \) - \( BC \) corresponds to \( bc \) ### Step 2: Write the ratios \[ \frac{AB}{ab} = \frac{AC}{ac} = \frac{BC}{bc} \] Substitute the values: \[ \frac{40}{x} = \frac{20}{14} = \frac{50}{y} \] ### Step 3: Solve for \( x \) Set \( \frac{40}{x} = \frac{20}{14} \): \[ \frac{40}{x} = \frac{20}{14} \] \[ 40 \times 14 = 20 \times x \] \[ 560 = 20x \] \[ x = \frac{560}{20} \] \[ x = 28 \] ### Step 4: Solve for \( y \) Set \( \frac{50}{y} = \frac{20}{14} \): \[ \frac{50}{y} = \frac{20}{14} \] \[ 50 \times 14 = 20 \times y \] \[ 700 = 20y \] \[ y = \frac{700}{20} \] \[ y = 35 \] --- ## **Final Answers:** \[ \boxed{x = 28} \] \[ \boxed{y = 35} \]