Given: [tex]$\triangle A B C$[/tex] is similar to [tex]$\triangle X Y C$[/tex].
If [tex]$B Y=4, Y C=7, X C=10$[/tex], which of the following proportions could be used to solve for [tex]$A C$[/tex]?
[tex]$4 / 11=10 / AC$[/tex]
[tex]$7 / 4=10 / A C$[/tex]
[tex]$7 / 11=10 / AC$[/tex]
[tex]$4 / 7=10 / AC$[/tex]
Answer (2)
Establish that corresponding sides of similar triangles are proportional.
Determine the length of BC by adding B Y and Y C : BC = B Y + Y C = 4 + 7 = 11 .
Set up the proportion BC Y C = A C XC , which gives 11 7 = A C 10 .
Conclude that the correct proportion to solve for A C is 11 7 = A C 10 .
Explanation
Analyze the problem and given information We are given that △ A BC ∼ △ X Y C . This means that the corresponding sides are proportional. We are given B Y = 4 , Y C = 7 , and XC = 10 . We want to find a proportion that can be used to solve for A C .
Set up the proportion Since △ A BC ∼ △ X Y C , we have the proportion BC Y C = A C XC . We know Y C = 7 and XC = 10 . We need to find BC .
Calculate BC We know that BC = B Y + Y C = 4 + 7 = 11 .
Substitute values into the proportion Now we can substitute the values into the proportion: 11 7 = A C 10 .
Identify the correct proportion Comparing this with the given options, we see that the correct proportion is 11 7 = A C 10 .
State the final answer Therefore, the proportion that could be used to solve for A C is 11 7 = A C 10 .
Examples
Understanding similar triangles and their proportional sides is crucial in various real-world applications. For instance, architects use these principles to create scale models of buildings, ensuring that the proportions are accurate. Similarly, mapmakers rely on similar triangles to represent large geographical areas on smaller maps, maintaining correct relative distances and sizes. By grasping these concepts, you can accurately scale objects and distances, making complex projects manageable and precise.
The correct proportion to solve for A C is 11 7 = A C 10 based on the similarity of the triangles and the lengths provided. This proportion was derived from the lengths of the sides of the similar triangles. Therefore, this is the option we choose.
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