Brief Summary
This video provides an introduction to similarity and size transformation in geometry. It covers the basic concepts of similar figures, enlargement, and reduction, and explains how to apply scale factors to find the dimensions and areas of transformed figures. The video includes step-by-step solutions to various problems, illustrating how to use formulas and scale factors in different scenarios, such as maps and models.
- Similar figures maintain the same shape but can differ in size.
- Size transformation involves enlargement (increasing size) or reduction (decreasing size).
- Scale factors are used to determine the ratio between corresponding sides of similar figures.
- Formulas are applied to calculate resulting lengths, areas, and volumes after size transformations.
Introduction to Similarity and Size Transformation
The chapter focuses on similarity as a size transformation, building upon previous concepts. Similarity in figures means they have the same shape, but their sizes can differ. For example, two triangles with proportional sides (e.g., 2:4, 3:6, 4:8) are similar. Similarly, circles of different radii are also similar because they share the same shape.
Similarity as a Size Transformation
Size transformation involves changing the size of a figure, either by enlargement (making it bigger) or reduction (making it smaller). Enlargement increases the size, while reduction decreases it. In size transformation, the original figure is the "object," and the transformed figure is the "image." Formulas are used to relate the resulting figure to the given figure using a scale factor (k).
Formulas and Scale Factor
The resulting figure is obtained by multiplying the given figure by a scale factor (k). For length, the formula is: Resulting Length = k * Given Length. For area, it is: Resulting Area = k^2 * Given Area. For volume, it is: Resulting Volume = k^3 * Given Volume. If k > 1, it's an enlargement; if k = 1, it's an identity transformation (no change); if k < 1, it's a reduction. In models and maps, the scale factor is often given as a ratio (e.g., 1:25000).
Question 1: Finding Sides of a Similar Triangle
Given a triangle ABC with sides 3.6 cm, 4.5 cm, and 6 cm, a similar triangle A'B'C' is created with the largest side (A'C') being 10 cm. The scale factor is found by dividing the new largest side by the old one (10/6 = 5/3). The lengths of the other sides of the new triangle are then calculated using this scale factor: A'B' = (5/3) * 3.6 = 6 cm and B'C' = (5/3) * 4.5 = 7.5 cm.
Question 2: Reduction of a Triangle
A triangle ABC has sides 16 cm, 12 cm, and 18 cm. A similar, smaller triangle A'B'C' is created such that the smallest side is 4.8 cm. The scale factor is calculated as 4.8/12 = 2/5. The lengths of the other sides of the reduced triangle are then found: A'C' = (2/5) * 16 = 6.4 cm and A'B' = (2/5) * 18 = 7.2 cm.
Question 3: Area of an Image After Reduction
A triangle has its size reduced by a scale factor of 0.72. If the original area of the triangle is 62.5 cm^2, the area of the image is calculated using the formula: Area of Image = k^2 * Given Area = (0.72)^2 * 62.5 = 32.4 cm^2.
Question 4: Finding the Scale Factor of Enlargement
A rectangle with an area of 60 cm^2 is enlarged. The area of the resulting image is 135 cm^2. To find the scale factor, the formula Area of Image = k^2 * Given Area is used. Thus, 135 = k^2 * 60, which gives k^2 = 135/60 = 9/4. Therefore, the scale factor k = √(9/4) = 3/2 = 1.5.
Question 5: Map Scales and Actual Lengths
On a map with a scale of 1:25000, a triangular plot has sides of 6 cm and 8 cm, with one angle being 90 degrees. Using the Pythagorean theorem, the third side is found to be 10 cm. The actual lengths of the sides are calculated using the scale. For the 8 cm side: Actual Length = 8 * 25000 cm = 2 km. For the 10 cm side: Actual Length = 10 * 25000 cm = 2.5 km. The area of the image is (1/2) * 8 * 6 = 24 cm^2. The actual area of the plot is then calculated, converting the map scale to square kilometers.
Question 6: Area Representation on a Map
The scale of a map is 1:200,000. The length on the ground represented by 1 cm on the map is calculated as 1 cm * 200,000 = 2 km. An area of 3 cm^2 on the map represents an actual area of 12 km^2 on the ground. The area on the map representing a 10 km^2 plot of land is calculated using the scale factor, resulting in 2.5 cm^2.
Question 7: Diagonal Distance on a Map
On a map with a scale of 1:200,000, a rectangular plot measures 32 cm by 24 cm. The diagonal distance on the map is found using the Pythagorean theorem to be 40 cm. The actual diagonal distance is calculated as 40 cm * 200,000 = 8 km. The actual area of the plot is calculated as 3.072 km^2.
Question 8: Dimensions and Volume of a Multi-Story Building Model
The dimensions of a multi-story building model are given, and the scale is 1/60. The actual dimensions are calculated: 1 meter * 60 = 60 meters, 60 cm * 60 = 36 meters, and 1.25 meters * 60 = 75 meters. The floor area of a room in the model is 250 cm^2, and the actual area is calculated as 90 m^2. The volume of a room in the model is related to the actual volume of 648 cubic meters, and the model volume is found to be 3000 cm^3.
Question 9: Ship Model Length, Area and Volume
A ship model is made to a scale of 1:250. If the length of the model is 1.2 meters, the actual length of the ship is 300 meters. If the area of the deck of the model is 1.6 m^2, the actual area of the ship's deck is 100,000 m^2. If the volume of the model is related to an actual volume of 1 km^3, the volume of the model is 64 m^3.
Question 10: Ship Model Dimensions and Volume
The length of a ship model is 4 meters, and the scale is 1:200. The actual length of the ship is 800 meters. If the area of the deck of the ship is 160,000 m^2, the area of the model's deck is 4 m^2. If the volume of the model is 200 liters, the actual volume of the ship is 16,000,000 m^3.
