Performing Indirect Measurement in Similar Figures
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Indirect measurement is a method of using proportions in similar figures to find a measure that is not easily measured directly. Here is a step-by-step guide:
Step 1: Identify the Similar Figures
The first step is to identify the similar figures in the problem. Similar figures are figures that have the same shape but may have different sizes. This implies that the corresponding angles are equal and the corresponding sides are in proportion.
Step 2: Set up the Proportion
Next, set up a proportion using the corresponding sides of the similar figures. The proportion is a ratio that equates two fractions. For example, if \(AB\) is a side in one figure and \(A'B'\) is the corresponding side in the similar figure, we can write the proportion as \(AB/A'B' = CD/C'D'\) where \(CD\) and \(C'D'\) are another pair of corresponding sides.
Step 3: Substitute the Known Measurements
Substitute the known measurements into the proportion. It is common that one of the measurements in the proportion is unknown. This unknown value is what you are solving for.
Step 4: Solve the Proportion
Solve the proportion to find the unknown value. This usually involves cross-multiplying and then dividing.
Step 5: Check the Solution
Finally, make sure to check your solution in the context of the problem. This is done to ensure that your answer makes sense.
Example
Let's consider a scenario where we need to find the height of a tree. We can use a meter stick and the shadow it casts to indirectly measure the tree's height.
Suppose a 1-meter stick casts a 2-meter shadow at the same time a tree casts a 10-meter shadow. Because the meter stick and the tree cast shadows at the same angle, we know that the triangles formed are similar. We can write the proportion like this:
\[ \frac{{\text{{Height of the tree}}}}{{\text{{Height of the stick}}}} = \frac{{\text{{Shadow of the tree}}}}{{\text{{Shadow of the stick}}}} \]
Plugging in the given values, we get:
\[ \frac{{\text{{Height of the tree}}}}{1} = \frac{10}{2} \]
By cross multiplying, we can solve for the height of the tree:
\[ \text{{Height of the tree}} = 1 \times \frac{10}{2} = 5 \, \text{{meters}} \]
So, the height of the tree is 5 meters.
