Question

an architect wants to build three similar triangles such that the ratio of the middle triangle. The smallest one has side lengths 5, 12, and 13; the largest triangle has the side lengths 45, 108, and 117. If two of the middle triangle's sides are 39 and 36, what is the length of the third side? ​

Answer 1

Answer:  15

Step-by-step explanation:

small:     5     12      13

middle:  __   36     39

large:    45   108    117

Notice that to get from the small to the large, multiplied by 9

5(9) = 45    12(9) = 108    13(9) = 117

That is because the sides are proportional $\dfrac{5}{45}=\dfrac{12}{108}=\dfrac{13}{117}\rightarrow\dfrac{1}{9}$

To get from small to middle, multiply by 3  

5(3) = 15     12(3) = 36    13(3) = 39

To solve it using proportions:

$\dfrac{5}{x}=\dfrac{12}{36}\\\\\\\text{Cross Multiply and solve for x:}\\5(36)=12x\\\\\dfrac{5(36)}{12}=x\\\\5(3)=x\\\\\large\boxed{15}=x$