Question

unit 05: triangle similarity benchmark test

complete the proof.
given: \\(ac = 6\\), \\(cb = 4\\), \\(ab = 7\\), \\(hi = 9\\), \\(hg = 6\\), \\(ig = 10.5\\)
prove: \\(\delta abc \sim \delta igh\\)

Explanation:

⚡ Using what you learned: similarity criteria for triangles (aa, sss, sas)

Step 1: Identify corresponding sides

Match the sides of \(\Delta ABC\) and \(\Delta IGH\) from shortest to longest:

• Shortest sides: \(CB = 4\) and \(HG = 6\)
• Medium sides: \(AC = 6\) and \(HI = 9\)
• Longest sides: \(AB = 7\) and \(IG = 10.5\)

Step 2: Calculate ratios of corresponding sides

Check if the ratios of the corresponding side lengths are equal:

$$ \frac{CB}{HG} = \frac{4}{6} = \frac{2}{3} \approx 0.67 $$

$$ \frac{AC}{HI} = \frac{6}{9} = \frac{2}{3} \approx 0.67 $$

$$ \frac{AB}{IG} = \frac{7}{10.5} = \frac{7}{\frac{21}{2}} = \frac{14}{21} = \frac{2}{3} \approx 0.67 $$

Step 3: Apply similarity criterion

Since all three pairs of corresponding sides are proportional:

$$ \frac{CB}{HG} = \frac{AC}{HI} = \frac{AB}{IG} = \frac{2}{3} $$

The triangles are similar by the Side-Side-Side (SSS) Similarity Theorem.

Answer:

The triangles \(\Delta ABC\) and \(\Delta IGH\) are similar by the SSS (Side-Side-Side) Similarity Theorem because the ratios of their corresponding sides are all equal to \(\frac{2}{3}\) (or \(1.5\) if calculated as \(\Delta IGH\) to \(\Delta ABC\)):

$$ \frac{IG}{AB} = \frac{HI}{AC} = \frac{HG}{CB} = 1.5 $$