Question

are these shapes similar?

Explanation:

Step1: Calculate the ratios of corresponding sides

For triangle \(GHI\) and \(UVT\), assume \(\angle G\) corresponds to \(\angle U\).
The ratio of the sides: \(\frac{27}{45}=\frac{3}{5}\), \(\frac{12}{20}=\frac{3}{5}\), but \(GH = 27\) and \(UT=45\), \(\frac{27}{45}=\frac{3}{5}\), while \(GI = 27\) and \(UV = 45\), \(\frac{27}{45}=\frac{3}{5}\), but in a triangle, for similarity, all corresponding sides must be in proportion. Here, if we consider the side - side - side similarity criterion.
Let's check all side - side ratios:
If we assume a correspondence (say \(GH\) with \(UT\), \(GI\) with \(UV\), \(HI\) with \(VT\))
\(\frac{GH}{UT}=\frac{27}{45}=\frac{3}{5}\), \(\frac{GI}{UV}=\frac{27}{45}=\frac{3}{5}\), \(\frac{HI}{VT}=\frac{12}{20}=\frac{3}{5}\)

Step2: Check the angle - angle similarity (since if sides are in proportion, angles are equal)

Since the sides are in proportion (\(\frac{3}{5}\) for all corresponding sides), by the Side - Side - Side (SSS) similarity criterion, the two triangles are similar.

Answer:

yes