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.
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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. (AB)/(DE) = (BC)/(EF) 6\, cmDE = 8\, cm12\, cm Step 3: Solve for DE. 6 × 12 = 8 × DE 72 = 8 × DE DE = (72)/(8) DE = 9\, cm The length of side DE is 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. (S_s)/(S_l) = (3)/(5) Step 2: Substitute the given side of the smaller triangle. Given S_s = 9\, cm. 9\, cmS_l = (3)/(5) Step 3: Solve for S_l. 9 × 5 = 3 × S_l 45 = 3 × S_l S_l = (45)/(3) S_l = 15\, cm The corresponding side of the larger triangle is 15 cm. Question 23: Step 1: Identify corresponding sides and the given ratio. The triangles XYZ and 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. (XY)/(DE) = (3)/(12) Step 2: Substitute the given length of XY. Given XY = 8\, cm. 8\, cmDE = (3)/(12) Step 3: Solve for DE. 8 × 12 = 3 × DE 96 = 3 × DE DE = (96)/(3) DE = 32\, cm The length of side DE is 32 cm. That's 2 down. 3 left today — send the next one.