31) If \( \triangle A B C \sim \triangle D E F \) and if \( A B=x^{2}-6 x-40 \) and \( D E=x^{2}-100 \), then what is the scale factor (in simplest form) of these similar triangles?
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To find the scale factor between the triangles, we need to set up the ratios of the corresponding sides. Given that \( AB = x^2 - 6x - 40 \) and \( DE = x^2 - 100 \), we can express the scale factor \( k \) as: \[ k = \frac{AB}{DE} = \frac{x^2 - 6x - 40}{x^2 - 100} \] Next, we can factor both expressions. The quadratic \( AB \) can be factored as follows: \[ x^2 - 6x - 40 = (x - 10)(x + 4) \] While the quadratic \( DE \) factors into: \[ x^2 - 100 = (x - 10)(x + 10) \] Now, substituting these factored forms back into our scale factor expression gives: \[ k = \frac{(x - 10)(x + 4)}{(x - 10)(x + 10)} \] Assuming \( x \neq 10 \) (since we can't divide by zero), the \( (x - 10) \) terms cancel out: \[ k = \frac{x + 4}{x + 10} \] Thus, the scale factor \( k \) in simplest form is: \[ \frac{x + 4}{x + 10} \] This represents the ratio of the lengths of the two triangles based on the side lengths given.
