To find missing dimensions in similar figures using a scale factor, you multiply a known dimension in one figure by the scale factor to find the corresponding dimension in the other figure, or divide if scaling down.
Understanding Similar Figures and Scale Factors
Similar figures are figures that have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. The scale factor is the ratio of the lengths of corresponding sides of similar figures. It tells you how many times larger or smaller one figure is compared to the other.
As shown in the provided video snippet, if you have a dimension of 6 in a smaller figure (DF) and the corresponding dimension in a larger similar figure (ABC) is 12, you can find the scale factor from the smaller to the larger figure:
- Reference Insight: "6 * 2 = 12. So the scale factor from DF to AB C from like small to big is two". This demonstrates that you multiply the smaller dimension (6) by the scale factor (2) to get the larger dimension (12).
Using the Scale Factor to Find Missing Dimensions
The process depends on whether you are scaling from the smaller figure to the larger figure or vice-versa.
Scaling Up (Small to Big)
When you know the dimensions of the smaller figure and want to find the corresponding dimensions in the larger similar figure, you use the scale factor as a multiplier.
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Formula: Missing Dimension (Big Figure) = Known Dimension (Small Figure) × Scale Factor
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Example: If the scale factor from figure A (small) to figure B (big) is 3, and a side in figure A is 5 cm, the corresponding side in figure B would be 5 cm × 3 = 15 cm.
Scaling Down (Big to Small)
When you know the dimensions of the larger figure and want to find the corresponding dimensions in the smaller similar figure, you divide by the scale factor.
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Formula: Missing Dimension (Small Figure) = Known Dimension (Big Figure) ÷ Scale Factor
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Example: Using the same figures, if a side in figure B is 15 cm and the scale factor from A to B is 3 (meaning the scale factor from B to A is 1/3 or dividing by 3), the corresponding side in figure A would be 15 cm ÷ 3 = 5 cm.
Steps to Find Missing Dimensions
- Identify Corresponding Sides: Determine which sides in the two similar figures correspond to each other.
- Calculate the Scale Factor: Find a pair of corresponding sides where both lengths are known. Divide the length of the side in the target figure by the length of the corresponding side in the original figure.
- Scale Factor (Original to Target) = Length of Side in Target Figure / Length of Corresponding Side in Original Figure
- Apply the Scale Factor: Use the calculated scale factor to find the missing dimension.
- If going from original to target (scaling up), multiply the known dimension in the original figure by the scale factor.
- If going from original to target (scaling down), divide the known dimension in the original figure by the scale factor (or multiply by the reciprocal of the scale factor).
Practical Application Example
Consider two similar triangles, Triangle XYZ and Triangle PQR. Side XY corresponds to side PQ, YZ to QR, and XZ to PR.
| Side | Triangle XYZ (Original) | Triangle PQR (Similar) |
|---|---|---|
| XY/PQ | 4 cm | 8 cm |
| YZ/QR | 6 cm | ? |
| XZ/PR | ? | 10 cm |
- Corresponding Sides: XY corresponds to PQ (4 cm and 8 cm are known).
- Calculate Scale Factor (XYZ to PQR): Scale Factor = Length of PQ / Length of XY = 8 cm / 4 cm = 2.
- This means Triangle PQR is 2 times larger than Triangle XYZ. This aligns with the concept from the reference: scaling from small (XYZ) to big (PQR) uses a scale factor greater than 1.
- Apply Scale Factor:
- Find missing side QR (in the big triangle): QR = YZ × Scale Factor = 6 cm × 2 = 12 cm.
- Find missing side XZ (in the small triangle): XZ = PR ÷ Scale Factor = 10 cm ÷ 2 = 5 cm.
By using the scale factor, you can efficiently determine any missing dimension in similar figures once the factor is known.
