How to find the missing length of similar triangles?
Understand the Problem
The question is asking how to calculate the missing length in similar triangles. To solve this, we will use the properties of similar triangles, which state that the ratios of corresponding sides are equal. This means we can set up a proportion based on the known lengths and use it to find the missing length.
Answer
The missing length is found using $b_2 = \frac{a_2 \cdot b_1}{a_1}$.
Answer for screen readers
The length of the missing side is calculated using the proportion of the similar triangles.
Steps to Solve
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Identify the sides of the triangles
Determine which sides correspond to each other in the two similar triangles. -
Set up the proportion
If the lengths of the corresponding sides of the triangles are known, we can set up a proportion. For example, if triangle A has sides $a_1, a_2$ and triangle B has sides $b_1, b_2$, the proportion can be written as: $$ \frac{a_1}{b_1} = \frac{a_2}{b_2} $$ -
Substitute known values
Plug in the known values into the proportion to solve for the missing length (let's say the missing length is $b_2$). -
Cross-multiply
Cross-multiplying helps eliminate the fractions: $$ a_1 \cdot b_2 = a_2 \cdot b_1 $$ -
Isolate the variable
Rearrange the equation to solve for the unknown length: $$ b_2 = \frac{a_2 \cdot b_1}{a_1} $$ -
Calculate the missing length
Insert the known values into the formula to find the missing length.
The length of the missing side is calculated using the proportion of the similar triangles.
More Information
Using the properties of similar triangles is a powerful tool in geometry which finds applications not just in theoretical concepts, but in real-world situations such as architecture and engineering.
Tips
- Forgetting to ensure the sides correspond to the correct triangles. Always double-check the triangles and what sides you're comparing.
- Not simplifying the proportion correctly; ensure that you perform the arithmetic operations accurately.
