Question

triangle abc is similar to triangle def with a side of triangle abc that is 58% of its corresponding side in triangle def. also, triangle ghi is similar to triangle def with a side of triangle ghi that is 45% of its corresponding side in triangle def
a. are \\( \triangle a b c \\) and \\( \triangle g h i \\) similar to each other? why or why not?
b. what is the ratio of corresponding sides of \\( \triangle a b c \\) and \\( \triangle g h i \\) ?
a. choose the correct answer below
\\( \bigcirc \\) a. \\( \triangle a b c \\) and \\( \triangle g h i \\) are similar by aa and sss similarity, but not sas similarity
\\( \bigcirc \\) b. \\( \triangle a b c \\) and \\( \triangle g h i \\) are not similar because not all corresponding angles in these triangles are congruent and/or not all corresponding sides in these triangles are proportional
\\( \bigcirc \\) c. \\( \triangle a b c \\) and \\( \triangle g h i \\) are similar by aa and sas similarity, but not sss similarity
\\( \bigcirc \\) d. \\( \triangle a b c \\) and \\( \triangle g h i \\) are similar by sas and sss similarity, but not aa similarity
\\( \bigcirc \\) e. \\( \triangle a b c \\) and \\( \triangle g h i \\) are similar by aa, sas, and sss similarity

Explanation:

Step1: Recall the transitive property of similar triangles

If \(\triangle ABC\sim\triangle DEF\) and \(\triangle GHI\sim\triangle DEF\), then \(\triangle ABC\sim\triangle GHI\) by the transitive property. Also, for similar triangles, all corresponding angles are congruent (AA similarity: if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar). Since \(\triangle ABC\sim\triangle DEF\) and \(\triangle GHI\sim\triangle DEF\), \(\angle A=\angle D=\angle G\), \(\angle B=\angle E=\angle H\), \(\angle C=\angle F=\angle I\). For SAS similarity: if the ratio of two sides of one triangle is equal to the ratio of two sides of another triangle and the included angles are congruent. Let the side of \(\triangle ABC\) be \(a = 0.58x\) (where \(x\) is the corresponding side of \(\triangle DEF\)) and the side of \(\triangle GHI\) be \(g=0.45x\). The ratios of corresponding sides of \(\triangle ABC\) and \(\triangle GHI\) will be proportional. For SSS similarity: if the ratios of all three pairs of corresponding sides of two triangles are equal. Since \(\frac{AB}{GH}=\frac{BC}{HI}=\frac{AC}{GI}\) (because \(\frac{AB}{DE}=\frac{GH}{DE}\) implies \(\frac{AB}{GH}\) is a constant, same for other sides), SSS similarity holds.

Step2: Calculate the ratio of corresponding sides

Let the side of \(\triangle ABC\) be \(s_{ABC}\) and the side of \(\triangle DEF\) be \(s_{DEF}\), \(s_{ABC}=0.58s_{DEF}\). Let the side of \(\triangle GHI\) be \(s_{GHI}\), \(s_{GHI} = 0.45s_{DEF}\). Then the ratio of corresponding sides of \(\triangle ABC\) and \(\triangle GHI\) is \(\frac{s_{ABC}}{s_{GHI}}=\frac{0.58s_{DEF}}{0.45s_{DEF}}=\frac{58}{45}\)

Answer:

a. E. \(\triangle ABC\) and \(\triangle GHI\) are similar by AA, SAS, and SSS similarity
b. The ratio of corresponding sides of \(\triangle ABC\) and \(\triangle GHI\) is \(\frac{58}{45}\)