This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) To find the scale factor of reduction from Triangle ABC to Triangle DEF, we divide a side length of Triangle DEF by the corresponding side length of Triangle ABC. Step 1: Identify corresponding sides and calculate the scale factor. The side AB (12 cm) in Triangle ABC corresponds to side DE (8 cm) in Triangle DEF. Scale factor of reduction = Length of DELength of AB = 8 cm12 cm Scale factor of reduction = (8)/(12) = (2)/(3) The scale factor of reduction is (2)/(3). b) Since the triangles are similar, the ratio of their corresponding sides is equal to the scale factor of reduction. Step 1: Calculate the value of y (EF). The side BC (9 cm) in Triangle ABC corresponds to side EF (y cm) in Triangle DEF. EFBC = Scale factor (y)/(9 cm) = (2)/(3) y = (2)/(3) × 9 cm y = 2 × 3 cm y = 6 cm Step 2: Calculate the value of z (DF). The side AC (15 cm) in Triangle ABC corresponds to side DF (z cm) in Triangle DEF. DFAC = Scale factor (z)/(15 cm) = (2)/(3) z = (2)/(3) × 15 cm z = 2 × 5 cm z = 10 cm The value of y is 6 cm and the value of z is 10 cm. c) The ratio of the areas of two similar triangles is equal to the square of their scale factor. Step 1: Use the scale factor from part (a) to find the area ratio. Ratio of areas = (Scale factor)^2 Ratio of areas = ((2)/(3))^2 Ratio of areas = (2^2)/(3^2) = (4)/(9) The ratio of the area of Triangle DEF to the area of Triangle ABC is (4)/(9). That's 2 down. 3 left today — send the next one.