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 the problems: 5. What is true about the angles of similar triangles? The corresponding angles of similar triangles are equal. Answer: The corresponding angles are equal. 6. Triangle A is similar to Triangle B. If the scale factor of A to B is 4 to 5, what is the ratio of the perimeters of A to B? For similar figures, the ratio of their perimeters is equal to the scale factor. Step 1: Identify the given scale factor. Scale factor of A to B is (4)/(5). Step 2: State the ratio of perimeters. The ratio of the perimeters of A to B is also (4)/(5). Answer: The ratio of the perimeters of A to B is (4)/(5). 7. ABC is similar to DEF. Find the perimeter of DEF. Step 1: Calculate the perimeter of ABC. Perimeter of ABC = AB + BC + AC = 8 + 10 + 12 = 30 cm. Step 2: Determine the scale factor from ABC to DEF. We compare corresponding sides. Side AC corresponds to side DF, and side AB corresponds to side DE. Using sides AB and DE: Scale factor k = (DE)/(AB) = (24)/(8) = 3. Using sides AC and DF: Scale factor k = (DF)/(AC) = (36)/(12) = 3. (Assuming DF=36 based on the scale factor, as DF is not explicitly labeled but DE=24 is). Step 3: Calculate the perimeter of DEF. Since the triangles are similar, the ratio of their perimeters is equal to the scale factor. Perimeter of DEF = Scale factor × Perimeter of ABC Perimeter of DEF = 3 × 30 cm Perimeter of DEF = 90 cm Answer: The perimeter of DEF is 90 cm. 8. A cube has all sides of length 2 cm. What will be the area of the hexagon be, if it is enlarged with scale factor 3? This question appears to have a slight mix-up in wording, referring to a cube but then asking about a hexagon. Based on the image, we will find the area of the enlarged hexagon. Step 1: Identify the original area and the scale factor. Original area of the hexagon = 60 cm^2. Scale factor k = 3. Step 2: Use the relationship between scale factor and area. If a figure is enlarged by a scale factor k, its area is enlarged by a factor of k^2. New Area = Original Area × k^2 New Area = 60 cm^2 × (3)^2 New Area = 60 cm^2 × 9 New Area = 540 cm^2 Answer: The area of the enlarged hexagon will be 540 cm^2. 9. What scale factor is needed to enlarge the smaller cuboid to become larger? To find the scale factor, we compare the lengths of corresponding sides of the two cuboids. Step 1: Identify corresponding dimensions. Smaller cuboid: Length = 4 cm, Width = 3 cm, Height = 2 cm. Larger cuboid: Length = 8 cm, Width = 6 cm, Height = 4 cm. Step 2: Calculate the ratio of corresponding dimensions. Ratio of lengths: 8 cm4 cm = 2 Ratio of widths: 6 cm3 cm = 2 Ratio of heights: 4 cm2 cm = 2 Since all ratios are equal, the scale factor is 2. Answer: The scale factor is 2.