Here are the solutions to the problems: 24. Two triangles are similar with sides in the ratio 4:5. If the area of the smaller triangle is $64 \text{ cm}^2$, find the area of the larger triangle. Step 1: Understand the relationship between the ratio of sides and the ratio of areas for similar figures. If the ratio of corresponding sides of two similar triangles is $a:b$, then the ratio of their areas is $a^2:b^2$. Given side ratio (smaller to larger) is $4:5$. So, the area ratio (smaller to larger) is $4^2:5^2 = 16:25$. Step 2: Set up a proportion using the area ratio and the given area. Let $A_s$ be the area of the smaller triangle and $A_l$ be the area of the larger triangle. $$ \frac{A_s}{A_l} = \frac{16}{25} $$ Given $A_s = 64 \text{ cm}^2$. $$ \frac{64 \text{ cm}^2}{A_l} = \frac{16}{25} $$ Step 3: Solve for $A_l$. $$ 16 \times A_l = 64 \text{ cm}^2 \times 25 $$ $$ A_l = \frac{64 \times 25}{16} \text{ cm}^2 $$ $$ A_l = 4 \times 25 \text{ cm}^2 $$ $$ A_l = \boxed{\text{}100 \text{ cm}^2\text{}} $$ 25. Define the term Circle and state the relationship between radius and diameter. A circle is a set of all points in a plane that are equidistant from a fixed central point. The relationship between radius ($r$) and diameter ($d$) is that the diameter is twice the radius, or $d = 2r$. 26. Find the circumference of a circle with a radius of $7 \text{ cm}$. ($\pi = \frac{22}{7}$) Step 1: Use the formula for the circumference of a circle. $$ C = 2\pi r $$ Step 2: Substitute the given values $r = 7 \text{ cm}$ and $\pi = \frac{22}{7}$. $$ C = 2 \times \frac{22}{7} \times 7 \text{ cm} $$ $$ C = 2 \times 22 \text{ cm} $$ $$ C = \boxed{\text{}44 \text{ cm}\text{}} $$ 27. Calculate the area of the following use $\pi = \frac{22}{7}$ a) Circle with diameter 14cm* Step 1: Find the radius. $$ r = \frac{\text{diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} $$ Step 2: Use the formula for the area of a circle. $$ A = \pi r^2 $$ Step 3: Substitute the values. $$ A = \frac{22}{7} \times (7 \text{ cm})^2 $$ $$ A = \frac{22}{7} \times 49 \text{ cm}^2 $$ $$ A = 22 \times 7 \text{ cm}^2 $$ $$ A = \boxed{\text{}154 \text{ cm}^2\text{}} $$ b) Circle with diameter 12cm* Step 1: Find the radius. $$ r = \frac{\text{diameter}}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm} $$ Step 2: Use the formula for the area of a circle. $$ A = \pi r^2 $$ Step 3: Substitute the values. $$ A = \frac{22}{7} \times (6 \text{ cm})^2 $$ $$ A = \frac{22}{7} \times 36 \text{ cm}^2 $$ $$ A = \frac{792}{7} \text{ cm}^2 $$ $$ A = \boxed{\text{}113.14 \text{ cm}^2 \text{ (approx.) or } \frac{792}{7} \text{ cm}^2\text{}} $$ c) Quarter circle with radius 10cm* Step 1: Identify the radius. $$ r = 10 \text{ cm} $$ Step 2: Use the formula for the area of a quarter circle. $$ A = \frac{1}{4} \pi r^2 $$ Step 3: Substitute the values. $$ A = \frac{1}{4} \times \frac{22}{7} \times (10 \text{ cm})^2 $$ $$ A = \frac{1}{4} \times \frac{22}{7} \times 100 \text{ cm}^2 $$ $$ A = \frac{2200}{28} \text{ cm}^2 $$ $$ A = \frac{550}{7} \text{ cm}^2 $$ $$ A = \boxed{\text{}78.57 \text{ cm}^2 \text{ (approx.) or } \frac{550}{7} \text{ cm}^2\text{}} $$ 28. The wheel of a bicycle has a radius of $35 \text{ cm}$. How many revolutions will it make in covering $1 \text{ km}$? Step 1: Calculate the circumference of the wheel. $$ C = 2\pi r $$ Using $\pi = \frac{22}{7}$ and $r = 35 \text{ cm}$: $$ C = 2 \times \frac{22}{7} \times 35 \text{ cm} $$ $$ C = 2 \times 22 \times 5 \text{ cm} $$ $$ C = 220 \text{ cm} $$ Step 2: Convert the total distance to centimeters. $$ 1 \text{ km} = 1000 \text{ m} $$ $$ 1000 \text{ m} = 1000 \times 100 \text{ cm} = 100000 \text{ cm} $$ Step 3: Calculate the number of revolutions. $$ \text{Number of Revolutions} = \frac{\text{Total Distance}}{\text{Circumference}} $$ $$ \text{Number of Revolutions} = \frac{100000 \text{ cm}}{220 \text{ cm}} $$ $$ \text{Number of Revolutions} = \frac{10000}{22} $$ $$ \text{Number of Revolutions} = \frac{5000}{11} $$ $$ \text{Number of Revolutions} = \boxed{\text{}454.55 \text{ (approx.) or } \frac{5000}{11}\text{}} $$ Since revolutions must be whole numbers for full rotations, the wheel makes 454 full revolutions.