Question

1. determine which two of the three triangles are similar. find the scale factor for the pair.

a.
b.

Explanation:

Step1: For part a

• In \(\triangle ABC\), using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), we find \(\angle B = 180^{\circ}-(38^{\circ}+ \angle A)\). Wait, no, better: In \(\triangle ABC\), \(\angle A=180^{\circ}-(38^{\circ}+ \angle B)\). Wait, for similarity, we can check side - ratios.
• For \(\triangle DEF\) and \(\triangle GHI\) (assuming \(I\) is the third vertex of the third triangle).
• Check the ratios of corresponding sides:
• \(\frac{DE}{GH}=\frac{4}{2} = 2\), \(\frac{DF}{GI}=\frac{7}{3.5}=2\), \(\frac{EF}{HI}\). Wait, no, for \(\triangle DEF\) sides \(DE = 4\), \(DF = 7\), \(EF = 6\); for \(\triangle GHI\) sides \(GH = 2\), \(GI=3.5\), \(HI = 3\).
• \(\frac{DE}{GH}=\frac{4}{2}=2\), \(\frac{DF}{GI}=\frac{7}{3.5} = 2\), \(\frac{EF}{HI}=\frac{6}{3}=2\).

Step2: For part b

• For \(\triangle ABC\) sides \(AB = 24\), \(BC = 16\), \(AC = 20\); for \(\triangle DEF\) sides \(DE = 18\), \(EF = 12\), \(DF = 16\); for \(\triangle GHJ\) sides \(GH = 30\), \(HJ = 20\), \(GJ = 25\).
• Check ratios:
• For \(\triangle ABC\) and \(\triangle GHJ\): \(\frac{AB}{GH}=\frac{24}{30}=\frac{4}{5}\), \(\frac{BC}{HJ}=\frac{16}{20}=\frac{4}{5}\), \(\frac{AC}{GJ}=\frac{20}{25}=\frac{4}{5}\).

Answer:

a. \(\triangle DEF\) and \(\triangle GHI\) are similar. Scale factor \(= 2\).
b. \(\triangle ABC\) and \(\triangle GHJ\) are similar. Scale factor \(=\frac{4}{5}\).