Question

solve for ( x ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

two triangles are shown: triangle ( abc ) with angles ( angle a = 46^circ ), ( angle b = 65^circ ), ( angle c = 69^circ ), side ( ac = 11 ), side ( ab = 12 ); triangle ( def ) (labeled as ( d ), ( e ), ( f )) with angles ( angle f = 69^circ ), ( angle e = 65^circ ), ( angle d = 46^circ ), side ( de = 31 ), side ( df = x )

answer attempt 1 out of 3
( x = ) input box submit answer

Explanation:

Step1: Identify Similar Triangles

Triangles \(ABC\) and \(DFE\) have the same angle measures (\(46^\circ\), \(69^\circ\), \(65^\circ\)), so they are similar by the AA (Angle - Angle) similarity criterion.

Step2: Set Up Proportion

In similar triangles, the ratios of corresponding sides are equal. For triangle \(ABC\) with sides \(AC = 11\), \(AB=12\) and triangle \(DFE\) with side \(DE = 31\) and \(DF=x\), we can set up the proportion \(\frac{AC}{DE}=\frac{AB}{DF}\) (since \(\angle A=\angle D = 46^\circ\), \(\angle C=\angle F=69^\circ\), \(\angle B=\angle E = 65^\circ\), so \(AC\) corresponds to \(DE\) and \(AB\) corresponds to \(DF\)).
Substituting the known values: \(\frac{11}{31}=\frac{12}{x}\)

Step3: Solve for \(x\)

Cross - multiply to get \(11x=12\times31\)
Calculate \(12\times31 = 372\), so \(11x = 372\)
Then \(x=\frac{372}{11}\approx33.8\) (rounded to the nearest tenth)

Answer:

\(x\approx33.8\)